How to construct the scheme below?

Question

How can one construct the diagram shown below, using just a ruler and a compass, where $ \angle PYX = 2\angle PXY $ and $ \angle PYZ = 2\angle PZY $:

Answer 1

The ratio doesn't seem to be anything special, but depends on the size of the angles in the triangle.

Let $M$ be the intersection of $YP$ and $XZ$. Using the cotangent formula or m-n rule, you can get the ratio $$\frac{XM}{MZ}=\frac{\sin ZPY\cos PXY}{\sin XPY\cos PZY}$$

Answer 2

A locus of points $P$ such that $\angle PYZ = 2 \angle PZY$ is a certain hyperbola.

A locus of poins $P$ such that $\angle PYX = 2\angle PXY$ is another hyperbola.

Finding an intersection of two hyperbolas is essentially a Problem of Apollonius. So the only thing left is to identify them.

Can you take it from here?