Last week Patrick Honner posed this clever geometry question on twitter:

There were many wonderful solutions posted on Twitter and on Google+, and I wanted to highlight 3 of them as wonderful examples of mathematical reasoning. These are the kinds of fantastic ideas that really highlight the beauty in math to me.

So, in the order that I saw them:

(1) Kate Nowak’s solution:

This solution uses two (related) ideas from geometry: similar triangles and ratios. There are two reasons that this solution was appealing to me. First, it is a fantastic use of abstraction – the point X that is part of the similar triangles and also part of the original triangle is not on the page, but we can still use properties involving these triangles in the solution.

Second, one of my great memories from learning geometry in Mrs. Whitney’s class (in, gulp, 1984 . . . ) was learning how to use ideas from geometric constructions to divide. Who knows why, but that idea just blew me away. In this solution, Nowak uses the ideas that Mrs. Whitney showed my class to divide a segment into two pieces of just the right proportion. The especially cool thing about Nowak’s construction is that we don’t know what the value of the ratio, only that the two line segments are divided into two pieces that are in exactly the same proportion. What a great, and super instructive, solution.

(2) Alexander Bogomolny’s construction:

I liked this solution because the approach is a clever twist on the ideas that Nowak’s used. Instead of dividing segments into equal, but unknown, proportions, Bogomolny divides two specific line segments in half. To achieve this goal he rotates a line about a point.

This gives rise to a nice question for a geometry student -> how do you do that with a compass and a straight edge?

Next he uses a neat property of trapezoids – the line connecting the midpoints of the two bases also passes through (i) the intersection of the diagonals, and (ii) the intersection of the two legs (when extended).

That give rise to a second great question for geometry students – prove that this statement about trapezoids (with non-parallel legs) is true!

So, I love the way that this proof not only answers the question posed by Patrick Honner, but also can be used to expose geometry students to a few other ideas and challenges.

(3) Patrick Honner’s solution:

Finally, the solution that Honner provided to his own problem is terrific:

The surprising (to me) idea Honner uses in this solution is that the altitudes of a triangle intersect at a single point. At first glance it is not at all clear how you might use this idea to solve the problem, but Honner shows how to find a triangle with X as vertex whose altitudes intersect at the point P. This solution shows a beautiful way in which two seemingly unrelated ideas in geometry are connected.

As I mentioned above, there were many solutions given on Twitter to Honner’s problem, but these three really stood out for me. I love that this problem illustrated how different people can approach a problem in different ways, and all of the different ways can be instructive.

Also, in a week when the internet spent way too much time discussing the differences between 5 + 5 + 5 and 3 + 3 + 3 + 3 + 3, it was nice to get this wonderful reminder from Nowak, Bogomolny, and Honner that math is about beautiful ideas and not just about getting an answer.

## Comments

Thanks for this post, Mike. I, too, really loved the variety of solutions. Craig Kaplan offered some great approaches on my G+ post (https://plus.google.com/u/0/+PatrickHonner/posts/9gmkhNrhCBX). In one, he essentially reconstructed the missing part of the diagram as a reflection, constructed the line of interest, and then reflected the solution back!

To be clear, I didn’t invent this problem. A mentor showed it (and a few others) to me many years ago, and I assume he learned of them through this Mathematics Teacher article from 1977: http://www.jstor.org/stable/27960971.

But given that so many people seem to enjoy them, maybe we should make some more!