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Looks Isosceles!

by Henri Picciotto

equiangular points (Logo)

Three unexpectedly related problems.

One day in the late 80's, I asked myself "if you were a bug in the plane, where would you have to stand for a given scalene triangle to appear to be isosceles?" In other words, given a triangle ABC, where are the points P such that (say) ∠APB=∠BPC? I called such points "equiangular points" for the given triangle, and investigated them with the help of a program I created in the Logo programming language.

I liked the figure I created in this process, and used it on the cover of my 1990 book Logo Math: Tools and Games. (See the figure above.) Notice that the locus of the points includes many parts. The easiest to understand are the straight-line extensions of each of the three sides, but it includes other curves. Also notice that in the figure above, there is a point inside the triangle, at the intersection of three curves, from where it appears equilateral.

In 1993, this problem (under the name "Henri's Problem") was discussed at an international gathering of academics about "Computers and Exploratory Learning" in Asilomar, CA. They used it as a way to compare different software tools (spreadsheets, the programming language Boxer, the interactive geometry program Cabri, and others.) Cabri was the clear winner of the comparison, in my opinion, as it allowed for a construction of the curves, and moreover revealed that the curves are cubics, as you can see in this figure:


However, note that on parts of the curve, the angles in question are supplementary, not equal -- a consequence of the way I constructed it. (If you have a better construction, let me know!)

Here is a manipulable figure: you can change the shape of the triangle by moving point A, and you can move P along the locus.

To manipulate points A and P, download the Cabri II Plus plug-in,
and use IE in Windows, or Safari on Mac OS X 10.4+.

At some point in the 90's, Jean-Marie Laborde (creator of Cabri) informed me that the problem had been solved by an Italian mathematician.

In July 2006, I attended the Park City Mathematics Institute, a rather wonderful gathering of math educators and mathematicians. Some interesting problems were posed to the secondary school teachers, which we were to solve with the help of interactive geometry software. Here are three of the problems. It was quite interesting to me to find that the second and third are in fact related to the equiangular points problem!

1. You're at point A, with an empty bucket. Your tent is on fire at point B. What is the shortest path from A to B if you want to stop at the straight river L to fill your bucket?

(You can move P to see what happens.)

This problem was familiar to me (I first encountered it many years ago in Chakerian, Stein, and Crabill's Geometry: A Guided Inquiry.) If you have not seen it before, you should definitely work on it.

2. Same problem, but this time you are on a circular island.

I don't have a proof or a simpler construction, but the optimal point in this case is an equiangular point on the circle, from which ∠APO=∠OPC. (It is easy to show that this is the point for which angle of incidence equals angle of reflection.) [More on this, and a Java applet, on Daniel J. Heath's site.]

best point

3. Finally: given a triangle ABC, where is the point P for which PA+PB+PC is minimum?

It turns out that this is the point mentioned above, where the triangle appears to be equilateral. This point is called the Steiner point. The figure below shows all the equiangular points. The Steiner point is at the intersection of all three curves inside the triangle.

all equiangular points

As it turns out, there is a construction of the Steiner point that does not use these loci, and which can be used as a foundation for the proof that this is indeed the optimal point. Can you find it?

Download the Cabri files: Tent-River, Tent-Island, Steiner, All Equiangular Points,
and the macro: Equiangular Points.
Visit Henri Picciotto's Math Education Page. Send me e-mail .