geometric extraversion

Geometric Extraversion

(Taken from my August, 2004 invited speech at the MAA MathFest)

In this talk I want to reintroduce you to two old friends, the incircle and the circumcircle of a triangle. I want to show you new features of these best known geometrical objects and illustrate new thoughts in triangle geometry introduced by John Conway. An algebraic version of this subject is here.

We begin by watching what happens to these two circles as we let angle B vary to zero. Play these movies one at a time by double clicking on them.

By letting angle B decrease to zero we rearrange the triangle so that the area goes to zero but the perimeter does not.

The area of the incircle goes to zero, but the that of the circumcircle does not. This simple fact indicates that the fundamental nature of these two objects is quite different.

If a variable goes to zero, it must be allowed to continue through zero if that is its wont.
As angle B and side b go to zero, we let them continue through zero.

The incircle becomes a circle outside ABC, but the circumcircle remains the same. Objects which remain the same are call "strong." Objects which change are "weak." Weak objects come in many versions; strong ones, only one.

We can understand this by adding some angle bisectors. Step 3 next.

To see how this happens, we reset the triangle and turn two of the angle bisectos on.

As the triangle varies, it becomes clear what happens, the red bisector, which began as internal, ends up being external. The same two bisectors which found the incenter, now find an excenter and so construct the excircle on the B side of the triangle.

Letting angle B go negative produces the B-excircle. Similarly we could produce the A-excircle or the C-excircle.

Hence the four in and ex-circles can be continously varied into eachother. We say the an excicle is an extra version of the incircle. Conway calls this new symmetry extraversion, which is a play on words, both meaning that an extra version is created by "extraverting" through a side.

Extraversion of Napoleon's theorem:

Napoleons theorem says that if equilateral triangles are erected outward on each side of ABC, their centers are connected to form an equilateral triangle.

Upon extraversion, the outward facing triangles become inward facing, producing an extra version of Napoleon's theorem.

Here we varied angle B; the same figure would result if we varied the other angles, hence there are only 2 version of Napoleon's theorem.

Step 6 next.

Another nice result, this one not as well known. The midpoint between an excenter and an incenter lies on the circumcircle. Here we show the example of the midpoint between Io and Ic, the incenter and the excenter on the C side. Symmetry would tell use that this theorem is likely to be true for both the Io, Ia and Io, Ib pairs, but extraversion tells us that it is true for all pairs.

The midpoints of the incentral quadrangle form a 6-fold object under extra-version. They also illustrate the powerful theorem that if one weak object (the midpoint) is on a strong object (circumcircle), all versions of the weak object are on that strong object.

By letting angle B decrease to zero we rearrange the triangle so that the area goes to zero but the perimeter does not. The area of the incircle goes to zero, but the that of the circumcircle does not. This simple fact indicates that the fundamental nature of these two objects is quite different.
If a variable goes to zero, it must be allowed to continue through zero if that is its wont. As angle B and side b go to zero, we let them continue through zero. The incircle becomes a circle outside ABC, but the circumcircle remains the same. Objects which remain the same are call "strong." Objects which change are "weak." Weak objects come in many versions; strong ones, only one. We can understand this by adding some angle bisectors. Step 3 next.

To see how this happens, we reset the triangle and turn two of the angle bisectos on. As the triangle varies, it becomes clear what happens, the red bisector, which began as internal, ends up being external. The same two bisectors which found the incenter, now find an excenter and so construct the excircle on the B side of the triangle. Letting angle B go negative produces the B-excircle. Similarly we could produce the A-excircle or the C-excircle.
Hence the four in and ex-circles can be continously varied into eachother. We say the an excicle is an extra version of the incircle. Conway calls this new symmetry extraversion, which is a play on words, both meaning that an extra version is created by "extraverting" through a side.


Extraversion of Napoleon's theorem: Napoleons theorem says that if equilateral triangles are erected outward on each side of ABC, their centers are connected to form an equilateral triangle. Upon extraversion, the outward facing triangles become inward facing, producing an extra version of Napoleon's theorem. Here we varied angle B; the same figure would result if we varied the other angles, hence there are only 2 version of Napoleon's theorem. Step 6 next.

Another nice result, this one not as well known. The midpoint between an excenter and an incenter lies on the circumcircle. Here we show the example of the midpoint between Io and Ic, the incenter and the excenter on the C side. Symmetry would tell use that this theorem is likely to be true for both the Io, Ia and Io, Ib pairs, but extraversion tells us that it is true for all pairs. The midpoints of the incentral quadrangle form a 6-fold object under extra-version. They also illustrate the powerful theorem that if one weak object (the midpoint) is on a strong object (circumcircle), all versions of the weak object are on that strong object.