[Math] Determining an angle inside the incenter of a triangle

Question

I've been working quite fruitlessly on a proof, with practically nothing to show for it. I feel as though I am missing something obvious, but nothing has come to mind, especially given the disproportionate amount of time I've spent on it.

So, suppose you have $\Delta XYZ$ with the incenter located at point $I$. Now, if we draw in segments $XI$ and $ZI$, we generate a smaller triangle $\Delta XIZ$. If we set $\angle Y$ to be something arbitrary, is there a way to determine what $\angle XIZ$ is? As in, some sort of general formula or relationship between $\angle Y$ and $\angle XIZ$?

I've tried some trickery using radii of the inscribed circle to try to get that angle, as well as using the center angles around $I$, but so far I haven't gotten anything out of it. Any ideas?

Thanks!

Answer 1

$\angle XIZ = 180^0 – \dfrac {X + Z}{2}$

Also, $(X + Z) = 180^0 – Y$

Result follows after eliminating X+ Z from the above.

Note that the location of I is fixed and comes directly from the triangle XZ and Y. It is not allowed to set Y as arbitrary after that.