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Geometry Of Midpoints

Tuesday, June 5, 2007

Last week a new puzzle titled Entrapment was posted on The Problem Site. This new puzzle has already become a big hit on the site, more than I anticipated. Within just a few days there were already close to 2000 puzzle solutions submitted.

On the puzzle page I gave the following hint:

Start out by trying some of the "Practice" puzzles. These are all puzzles which contain exactly three red and three gray dots. What you will discover is that most puzzle solutions are built around triangles, so if you can master the easy puzzles, the more challenging puzzles can be broken down into smaller, more manageable pieces.

I thought I would elaborate on that just a little bit, because the Geometry of the Entrapment Puzzle is quite interesting. The Geometry of Entrapment is the Geometry of Midpoints, and of Triangles.

Take a look at the following image, which shows three completed "Practice" puzzles:

What do you notice about each of these solutions? First, you should notice that the three red dots form a triangle (of course! Three points in a plane always determine a triangle!). Second, of course, the gray dots form a triangle as well. What should also be apparent is that the gray triangle is similar to the the red triangle. And when I say "similar", I'm using the geometric definition of the word similar. These two triangles have congruent angles, and equal ratios of sides.

In fact, if you think about it for a moment, you'll be able to figure out what that ratio of sides is. Are you thinking about it? If you're not sure, you can scroll down to the bottom of this post, where you'll find the answer.

Now, the other thing which is interesting and helpful is the fact that if you pick any two red dots and draw a segment connecting them, there will be two gray dots that define a segment parallel to that segment. Take a look at the picture if you don't believe me. Remember that from your Geometry class?

All of a sudden you should realize why the "Practice" puzzles are called "Practice" puzzles - once you understand the Geometry of the situation, there is very little Problem Solving involved; you can solve each one of the Practice puzzles by counting distances and creating parallel segments.

So why bother with the Practice puzzles? Because if you can get the hang of entrapping triangles, the other puzzles get much easier. Take a look at this puzzle:

A puzzle like this can look very intimidating - and some are even more intimidating than this. But if you can quickly locate triangles to entrap, the puzzle gets easier. Here's an example:

It took me only moments to recognize that triangle which could be entrapped. Note the similar triangle, the common ratio, and the parallel sides. From there I would start placing other dots to maximize the number of red dots entrapped. But it all starts with finding a triangle you can entrap.

When I first started solving these puzzles, I thought This is insane - these beastly things are unsolveable! In reality, even the most challenging Entrapment puzzles can be solved within a minute or two, if you can find the correct triangle to start with!

Have fun ENTRAPPING!

Ratio of Sides: Well? What is the ratio of sides? Did you say 2:1? If you did, you are correct. I won't take the time to do a proof here, but it's well known that a segment connecting midpoints of two sides of a triangle has half the length of the third side.

Labels: entrapment, geometry, theproblemsite

posted by Douglas Twitchell at

2 Comments:

I had fun using your midpoints puzzle. I would like to show it to my Geometry classes. It is a great tool for showing students about similarity and the rules of midpoints.

By Geometry Teacher, At June 28, 2007 1:06 PM

I'm glad you enjoy the puzzle. And I hope your students enjoy using it. I'd love to hear back from you about how well it worked in your class!

By Douglas Twitchell, At June 28, 2007 1:42 PM