If $XYZ$ is a right scalene triangle and if one of its constituent triangles is equilateral, then the other triangle must be isosceles. [closed]

$\begingroup$

This question is related to a component of a proof I constructed. I have posted the (pending) proof for verification on MSE: .

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.

Thank you.

$\endgroup$ 13

1 Answer

$\begingroup$

First question. If the constituent triange is equilateral then one of the angels of the right triangle is 60. And the other is 30. Removing the equilateral triangle you'll be left with a triangle with the 30 degree angle and the 90 degree angle less 60 degrees. As two of the angles are both 30, it is isoceles.

You second question is clearly false as only 60-30-90 triangles will have this property.

$\endgroup$ 3