Find the sides of a right triangle formed by connecting two other right triangles from the center of their hypotenuse.

Question

I have the following sketch of the problem:

I need to find the values of $x$ and $y$ in the previous drawing. The hypotenuses of both black triangles are of equal length and the red triangle is a right triangle constructed by connecting the centerpoint of both hypotenuses. So far I was able to find out how to calculate gamma:

$\gamma = \alpha+ (\beta- \alpha)/2$

But I am clueless on obtaining the actual value of $x$ or $y$ and could use some help with this.

Answer 1

Suppose the black triangles have hypotenuse 1.
The upper black triangle has horizontal side $\cos\alpha$, and the lower black triangle has horizontal side equal to $\cos\beta$.
The part of the red $x$ inside the lower triangle is half of $\cos\beta$, by similar triangles.
The part of $x$ outside the lower triangle is part of a rectangle, and the upper side of the rectangle is half of $\cos\alpha$.
So $x=\frac12(\cos\alpha+\cos\beta)$