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 with , given points on and on such that we have , 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 , let be the midpoint of and be on such that . Then, . 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)?