A synthetic approach to a Geometry problem

Can you find the value of angle $x$ in degrees without using any trigonometric identities (or) formulas. That is, you should not be using any algebra related to trigonometry to solve it. Try to build a synthetic (elegantly presentable) solution using some constructions before scrolling down.......

Need a hint ! Here are two...

First, draw a neat and large diagram and label all essential points.

Second, let the midpoint of the line segment bisected by the circle be $F$ . Extend $OF$ to meet the base of the right-angled triangle at $G$ . Find some congruent triangles. Now try to find $x$ by yourself.......

Here is a solution... First let's draw the diagram with $OF$ extended as shown,

$\text{Let the radius of incircle be }r.\\\text{Extend }OF\text{ to meet }AB\text{ at }G \Rightarrow OE\ ||\ GB\text{ and}\\\text{hence }\triangle O E F \cong \triangle G B F\ (\text{by }ASA),\\\Rightarrow \angle B F G=x \Rightarrow \angle F G A=2 x \text { and } O F=F G=r=OE=GB.\\\text { In } \triangle O D G,\ \sin (\angle O G D)=\frac{O D}{O G}=\frac{r}{2 r}=\frac{1}{2}\Rightarrow \angle OGD=30^{\circ}\\\Rightarrow 2 x=\angle F G A=\angle O G D=30^{\circ}\\\boxed{\Rightarrow x=15^{\circ}}$

I believe that there are many other solutions using synthetic method. Kindly post your ideas in the comments below.....

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A synthetic approach to a Geometry problem

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