What are some non-trivial ways to construct a similar triangle from a right triangle? By non-trivia

Schenone2pare

Schenone2pare

Answered question

2022-06-01

What are some non-trivial ways to construct a similar triangle from a right triangle?
By non-trivial, I mean not:
- The triangle itself or any translations or rotations of it
- A 'shrinking' of the right triangle; that is, for A B C with A B C = 90 , given points D on A C ¯ and E on B C ¯ such that D E C = 90 we have D E C A B C, along with any arbitrary renaming of points. Or, more generally, any scaling of the triangle.
One of the most well-known ways to construct a similar triangle is by dropping a cevian from the hypotenuse to the vertex which is perpendicular to the hypotenuse. In other words, an altitude.
I can only think of one more case, and I'm not sure if it is true (verification would be appreciated); given right triangle A B C, let D be the midpoint of A C ¯ and E be on B C ¯ such that B E D = 90 . Then, B E D A B C. In fact, I think this triangle is congruent to the one mentioned in the last paragraph. Also not sure on that, though.
The last one came about as a surprise while I was working on a proof, but I could not prove it nor could I think of any other ways to construct a similar right triangle to one given. So, my question: given a right triangle, what are some ways to construct a similar triangle (preferably ones that you found clever/interesting)?

Answer & Explanation

Nancy Sandoval

Nancy Sandoval

Beginner2022-06-02Added 2 answers

Let M, N, P be the middle of A B, B C, and C A. You have now 4 triangles similar to the initial one.

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