The Three
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We look at how to cross the Three / Four bridge geometrically: in both directions. This connects with some classical triangle geometry, involving triangle centres going back to ancient Greek geometry. We will touch base with the algebraic orientation to angle bisection, modifying a little the approach of Euclid in Proposition 9 of the Elements. And we'll discover the rather surprising connection between incentres and orthocentres.
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[Music]
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hello everyone i'm norman wildberger and
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welcome to six
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an elementary course in pure mathematics
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so we've been talking about the bridge
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between three and four
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that's kind of on our way to
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understanding six things
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and it will turn out that there's
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another bridge between two numbers
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a little bit bigger that will also play
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a very important role
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so we want to understand this 3-4 bridge
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quite well
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and this is really nice because it
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connects with a lot of other
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actually important things in mathematics
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and today we're talking about
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connections with geometry in particular
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triangle geometry
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and we're going to see connections
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between the classical in centers and
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ortho centers and this 3 4 bridge so let
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me remind you we start with four points
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say one two three four in the plane
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and then we create these diagonal points
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where opposite
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lines or edges meet and we get the
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points a b
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c forming the diagonal triangle so we
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are going from
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four points to three points all right
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now i should maybe say this is the
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diagram that i usually make for this but
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it's probably
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important to realize that these four
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points that we start with
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don't have to form a convex
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quadrilateral
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okay so it's in other words possible for
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one of the points to be
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inside the triangle formed by the other
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three
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so perhaps you might like to draw a
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corresponding picture
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like that where say the point 4 is moved
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from out here
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to say being inside the triangle 1 2
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- everything still works
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we just get a different looking
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configuration
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now the question that we want to address
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today is how do we go
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in the other direction on this bridge if
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we start with
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three things like say the triangle abc
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in the plane
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how can we go back to a quadrilateral or
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quadrangle of
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four points so that we still have this
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relationship
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now there's a bit of an asymmetry here
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because four is bigger than three
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and although it's reasonable to go from
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four points to three points uniquely
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like in this situation here it's not so
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obvious that we should be able to go
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backwards
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uniquely in other words we can't really
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expect that
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given three points there's a unique
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configuration of
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four general points that are somehow
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determined by them
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okay so that's a bit of an aspect here
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four to three we're losing some
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information
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and we can't recover it so quickly
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anyway but
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answering this question is quite
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interesting and connects with
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some interesting triangle geometry
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triangle geometry is the study of this
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very elementary configuration
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just three points abc and the three
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lines in the plane that they form
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it's a aspect of geometry that goes back
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to the ancient greeks
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who discovered and investigated the
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four simplest triangle centers the in
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center
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the centroid the circumcenter and the
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ortho center so these go
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back a thousands of years but in more
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modern times in european times
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a lot more additional triangle centers
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have been discovered
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first hundreds and then thousands of
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them and so now it's an incredibly
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rich topic
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perhaps the richest branch of
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mathematics
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actually so let's talk
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about the triangle center the in center
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first
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actually that's the first one on clark
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kimberling's
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famous list so he has a website
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called encyclopedia of triangle centers
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which
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lists thousands of triangle centers
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and the in center is actually the very
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first one on his list
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so this is a great resource for people
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interested in geometry and i encourage
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you to have a look at it
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so how do we get the in center well
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there it is there for this triangle here
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it's the point that has the property
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that you can
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inscribe a circle in the triangle
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with center at that point i
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so that's the center of this circle here
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which is inscribed in the triangle
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meaning that is
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tangent to the three sides of the
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triangle
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to construct it classically one looks at
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so-called angle bisectors
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which i prefer to call lines so for
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example
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at a this is an angle bisector or a
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byline
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traditionally it means that this angle
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here and this angle here are the same
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in rational trigonometry we prefer to
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think about spreads which are more
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algebraic and computational than angles
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so the corresponding statement would be
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that this spread is the same as this
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spread
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okay and then there's a corresponding
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angle bisector or
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by line at b and another one at c
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and lo and behold these three
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intersect at this point i and that's the
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in center
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and that's the very first point on clark
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kimberling's list of triangle centers
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euclid's famous book the elements is
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very interested in constructions
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geometrical constructions and
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constructions of angle bisectors or
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bylines
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figures prominently even in book one in
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proposition 9
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he shows how to construct bisectors or
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bylines
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when you have two lines meeting at a
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point a
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he says what you should do is you should
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draw a circle centered at a
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and see where it meets the two lines in
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question
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and then you connect those two points
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with a line like this
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and then you actually use proposition
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one of book one which shows how to
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construct an
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equilateral triangle on this base
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okay so what you do is you get your
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compass and you
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put the center here and you draw a
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circle through the other point
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so something like like this okay that's
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supposed to be a circle around
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this point here and then you move the
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compass to this point here and you draw
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a circle
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with center here through the other point
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okay that's some other thing
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some other circle like this all right
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and then where those two circles
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meet well they actually meet in two
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different places but say that would be
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one of them there
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okay and then you connect these points
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there and that's
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then an equilateral triangle
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so that's the very first proposition of
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book 1.
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of course then if you can do that well
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then you can join this
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new point that we're creating here with
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the original
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vertex a and then you get a angle
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bisector so that that spread is the same
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as that spread
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or if you're talking about angles the
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angles are equal
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as far as euclid would have stated it he
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would have said well this
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angle is equal to that angle in the
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following sense that this geometrical
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configuration is the same as this
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geometrical configuration in terms of
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picking this up and putting it down over
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there
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so he does not actually talk about angle
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measurement okay he talks about angles
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as geometrical objects that can be moved
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and superimposed on each other
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okay but in any case that's how the
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classical story
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of angle bisectors is uh usually
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explained
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now actually when we go to a more
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algebraic
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modern point of view we see that the
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situation is a little bit
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more subtle and actually involves some
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some number theoretic considerations
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which are maybe not so obvious when
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you're just
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drawing on a piece of paper so the real
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story is
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more like this you have a point there
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let's call it a as well
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and you have these two lines meeting but
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actually the lines go off in
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different directions right they're not
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just necessarily rays
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so how do we form angle bisectors for
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this configuration
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well we still use the circle center
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today and see where it meets
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say the two lines so we could you know
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use that point and that point
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and then we could do the construction
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that we've just done and get this
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equilateral triangle
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there and then we can say aha now we
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have an angle bisector
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fine but actually when you're drawing
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the circle
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there's some choices involved because
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this circle meets
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one of these lines if it does at all in
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two points typically
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so we could also have chosen say this
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point here and
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this point over here
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if we had done it that way then the
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equilateral triangle that we would form
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would be up here somewhere like this
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and then when we join with a we would
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get this
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as an angle bisector okay
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so actually if you do this sort of more
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general thing and look at all the
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possibilities
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you get two angle bisectors and
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crucially they
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are perpendicular okay
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so i might write that there so there are
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two
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bylines
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and they are perpendicular
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so that's interesting that allows us to
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think about
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in centers in a broader actually better
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way
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okay so let's return to that initial
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picture that i showed you of the in
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center
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and let's look at a little bit in a
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broader setting
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okay so here's a triangle a b c okay and
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when we are constructing angle bisectors
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or by lines now we see that they come in
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pairs
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so that's the first one that we saw but
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there's actually this other one which is
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in some sense algebraically just as
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valid
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a solution to what's ultimately a kind
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of a quadratic equation if you write
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down the algebra for it
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so we have these two by lines which are
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actually perpendicular
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at a we have these two by lines which
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are actually perpendicular at b
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and we have these two by lines which are
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actually perpendicular at c
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so starting with the triangle abc we
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actually have
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six by lines okay that makes us happy
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because we like six all right and then
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these six bylines meet in various places
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they meet at least three of them meet at
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this
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first in center that we've already
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discussed which is the center of this
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circle inscribed in the triangle but
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also
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the other ones also meet in three
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additional points
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so this by line and this spy line and
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this spy line
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meet in i2 and these three here meet in
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i3
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and these three here meet in i4
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and these other in centers sometimes
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they're called
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x centers but it's really better to
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think of them as all being in centers
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okay
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they are also sharing the property that
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they are the centers of circles
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which are tangent to the original sides
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of the triangle
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as you can see here so i2 is the center
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of this circle which is still
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tangent to the three sides of the
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triangle
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i3 is the center of this big circle here
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i4 is the center of this circle here
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so you see that what we're doing is
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we're crossing
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this 3 4 bridge because we're starting
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with a triangle
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and we're making a construction that
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ends up with
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4 in centers so we get this quadrangle
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i1 i2 i3
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i4 and
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if we look at the diagonal triangle of
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that how do we get that well we just
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join these various four points in all
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six
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possible ways and see what the new
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points of intersection
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are well this
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edge and this edge will meet in c
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this edge and this edge will meet in b
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this edge and this edge will meet in a
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so abc is the diagonal triangle
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for this in center
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quadrangle so we're going now from three
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to four and this is reversing the
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pattern that we saw
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originally going from four to three so
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now we can cross the three four bridge
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in either direction now i want to talk a
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little bit about the ortho center
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that's another famous triangle center
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which usually is quite separate in some
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sense from
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the in center story but we'll see that
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there's actually a very interesting
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connection
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having to do with this 3-4 bridge so if
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you have a triangle say
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a b c then the ortho center
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is the meat of the altitudes
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so an altitude is aligned through a
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vertex like this one here
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which is perpendicular to the opposite
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side
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side bc so that's an altitude the
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altitude through b is this line here
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it's perpendicular to ac
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and the altitude through c is this line
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here it's perpendicular to a b
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and as you can see here these three
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altitudes meet
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at a point d the fact that these three
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altitudes meet is actually
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quite non-trivial and
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folklore has it that it was albert
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einstein's favorite
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theorem in elementary geometry it is a
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very nice fact
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but when we look at this picture so d is
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the orthocenter of triangle abc
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we actually see that there's more
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symmetry around
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and we're interested in symmetry we also
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see that in fact
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the same diagram shows that the point a
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is actually the orthocenter of the
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triangle
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b c d
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let's think about that okay so if we had
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started with this triangle bcd
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what are the altitudes of it
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well suppose we're looking at the
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altitude from d
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to the opposite side so that would be
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this line here
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what about the altitude from b to the
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opposite side
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the opposite side is cd well the
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altitude line is going to be this line
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right here so that's an altitude
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and what about the altitude from c to
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this opposite side
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that's going to be this line here so the
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three altitudes
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of the triangle b c d are
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are this line this line and this line
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and lo and behold
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they do meet at the point a so
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a is the ortho center of the triangle b
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c
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d and similarly b
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is the ortho center of the triangle a c
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d
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and c this one here is the ortho center
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of the triangle a b d
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so when we just look at a diagram like
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this
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we actually can see four different ortho
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centers of
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four different triangles these four
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points
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a b c d have a very pleasant
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four-fold symmetry and
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we say that they are an orthocentric
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quadrangle so a b
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c d is an orthocentric quadrangle
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any one of those four points is the
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orthocenter
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of the other three
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now associated with this orthocentric
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quadrangle abcd
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is this triangle e f g
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what's that that is a triangle formed by
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the
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feet of the altitudes so if we go back
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to the original triangle abc
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and then we looked at the altitudes here
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the feet of the altitudes the places
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where those
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altitudes meet the opposite sides are
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the points e
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f and g so that triangle is the
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feat of the altitudes called the orthic
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triangle of
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abc but it
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it's easy to see just from the kind of
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thing i've been saying
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that it's also the ortho triangle of
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a b d or of
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b c d or of a c d
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so this is a kind of common orthic
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triangle
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to all of the
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triangles formed in this orthocentric
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quadrangle
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so it's kind of another 3 4 connection
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right we've got
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these four points a b c d forming this
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very special quadrangle
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and then this triangle efg which
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is somehow beautifully associated to the
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quadrangle
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going from four to three
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okay so now i can tell you a magical
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fact something which is not obvious
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something which you look at and say wow
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that's cool
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and that's typical of the beauty of
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triangle geometry
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it's that these two diagrams i've shown
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you these two pictures
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of the in center story of a triangle and
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these
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orthocentric quadrangles they are
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really in some sense the same picture
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okay so let me show you so here's the
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sort of the first picture that i
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started with we started with the
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triangle abc and we were looking at
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in centers formed by by lines at the
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at the various vertices okay and we had
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these four in centers
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i1 i2 i3 i4
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now if we just look at this quadrangle
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i1 i2 i3
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i4 that should be a1
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okay let's make it a one okay that thing
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is actually an
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orthocentric quadrangle
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and its orthic triangle is the original
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triangle abc
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so what makes it an orthocentric
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quadrangle well
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any one of these in centers is
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actually the orthocenter of the triangle
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formed by the other three
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in centers so for example this i1 here
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is actually the ortho center of the
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triangle
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i2 i3 i4
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and that's kind of obvious because we
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know
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that these bylines are perpendicular so
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this is a byline
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at b and this is a byline at b
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so yes indeed this line here is really
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the altitude from i3
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to i2i4
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this is an altitude from i2 to i3i4
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and uh and this one here is an altitude
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from i4 to i2 i3 so yes indeed
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i1 is actually the orthocenter of i2i3i4
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and then from that symmetry that we've
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already observed
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that means that this is an orthocentric
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quadrangle and
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similarly i2 is the orthocenter of
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i4 i1 etc
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so we can see the orthocentric
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quadrangle picture
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in the in center story
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but conversely if we go starting with
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the
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orthocentric quadrangle source so here's
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a copy of that
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ortho center diagram we started with a b
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c and we drew altitudes and we found
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d to be the orthocenter of the triangle
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abc and then we said
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okay now we actually have an
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orthocentric quadrangle abcd
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with orthic triangle efg
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what i'm claiming now is that this
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diagram here
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is actually really the same as an
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in-center
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diagram namely abcd
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this is actually the quadrangle of in
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centers
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of d e f
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this triangle d e f here this orthic
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triangle its in centers
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are exactly d a
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b and c
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that's a non-trivial fact it's not so
20:56
easy to prove
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if you have some experience in geometry
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you might like to try to
21:00
prove that but for your homework i don't
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want you to prove it
21:04
i want you to verify it so i'm making
21:08
statements here i want you to
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open up geogebra and test this out and
21:13
see is this really true
21:15
so if we start with a triangle abc and
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we construct
21:18
its orthocenter and then construct this
21:20
orthic triangle here
21:21
is it really true that these things
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are in centers in other words is this
21:27
really
21:28
a byline or an angle bisector of the
21:31
triangle efg
21:34
you can verify that using geogebra
21:38
okay so that'll make you
21:41
confident that yes these statements are
21:44
probably true
21:45
even if they do have a kind of a magical
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aspect
21:49
all right so in today's lecture we've
21:52
shown that this 3-4
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bridge actually has a manifestation in
21:57
the world of triangle geometry
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and it connects naturally with the four
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in centers
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of a triangle and it connects with the
22:07
orthocentric quadrangle that we get by
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looking at ortho centers of a triangle
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and that these stories are in some sense
22:16
really the same
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so this three-four bridge is quite rich
22:21
i'm norman wahlberger thanks for
22:24
listening
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[Music]
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you