## My Equilateral Triangle Puzzle (solutions)

Solutions (and a couple of additional questions):

1. Is it possible to divide an equilateral triangle into two equal (same area) triangles, which are not congruent to each other? Of course not, the diagram shows the only way to dissect an equilateral triangle into two equal triangles, and they are congruent.

2. Divide an equilateral triangle into three equal triangles, none of which is congruent to any of the others. The diagram on the right (or some equivalent reflection or rotation) shows the only two ways.

3. Divide an equilateral triangle into four equal triangles, none of which is congruent to any of the others. On the left are six ways to do it. I have labeled two of them A and B, to help with the solution of question #4. Let me pose a fifth question: Question #5 - Is there any other way to divide our equilateral triangle into four equal triangles with none of them being congruent? The answer to this question is below.

4. Divide an equilateral triangle into five equal triangles, none of which is congruent to any of the others. On the right are two ways, similar to A and B in the solution #3. Of course, there are many other solutions.

5. Here is a seventh solution to question #3. I think that there is no eighth solution.

Let me ask a question #6: Do the two methods in questions 3 and 4, labeled A and B, always work? In other words, can we divide an equilateral triangle into n equal triangles, using those two methods, no two triangles being congruent? The answer to that is below.

6. Yes, both methods work for all integers n. The only difficulty is proving that no two triangles are congruent. You might want to think about that. Examining method A, we see that the smallest angle of each triangle (labeled a, b, c, . . . on the right) follows this relationship: a<b<c<d<e< . . . In other words, none of the smallest angles is equal, and the triangles are not congruent. The proof may or may not be easy?

Examining method B, we see that two triangles (or more) are congruent only when one of the sides is vertical (first diagram on the left), or if one of the triangles is isosceles (second diagram on the left). I think the Pythagorean theorem will show that the colored triangle in each diagram cannot have the correct area.