Find the value of the angle $X$ in the given figure

Question

In this picture, the curve in the inside of the big outer triangle is actually its incircle. The edges of the triangle inside the incircle are the intersections of the incircle with the outer triangle.

What is the value of angle $X$ in the given figure?

Answer 1

Let's use the notation as in this picture.

Note that $\overline{DE} \perp \overline{BC}$ and $\overline{DG} \perp \overline{AC}$. Hence $\angle GDE= 180-\angle EBG$. AS the traingle $\triangle GED$ is isosceles, we conclude $$\angle DEG= \frac{1}{2}( 180-\angle GDE)= \frac{1}{2} \angle EBG$$ Similarly $$\angle FED= \frac{1}{2} \angle FCE$$

In conclusion $$\angle FEG =\frac{1}{2} (\angle FCE + \angle EBG)$$

Answer 2

Let's call $a,b,c$, ($b$ is opposite to $x$ and $a$ is the up arc) those three arcs on the circle:

$$\frac{a+b-c}{2}=32$$ $$\frac{b+c-a}{2}=84$$

Sum both equations and get $b=32+84$ and then $x=b/2=58$

Answer 3

HINT

It is an in-circle of the given triangle.Formed by corner angle bisectors. Angle chasing can find it.