This question is related to a component of a proof I constructed. I have posted the (pending) proof for verification on MSE: https://math.stackexchange.com/questions/2067151/use-the-information-given-to-construct-a-proof.
However, I would greatly appreciate it if someone could verify whether this specific statement I made is true:
If $XYZ$ is a right scalene triangle and if one of its constituent triangles is equilateral, then the other triangle must be isosceles (and vice-versa).
In the above proof, this statement was made to justify the following key question and abstract answer:
How can I show that a right scalene triangle consists of an equilateral triangle and an isosceles triangle?
Show that one of the triangles is an equilateral triangle.
In the above question, I also included the following diagram to illustrate the triangles:
Notice that the right scalene triangle consists of two other triangles when divided.
If anything is unclear, you may find clarification in the proof question posted above. Also, I will be glad to answer any questions if you'd prefer to ask me directly.
If my reasoning is incorrect, please explain why.