# Finding the area of a triangle, given the distance between center of incircle and circumscribed circle

Consider the following depiction:

$ABC$ is an isosceles triangle ($AB=AC$), where the two angles opposite the equal sides are equal $\beta$ ($\beta>60$), and $AD$ perpendicular to $BC$.

$O$ is the center of the circumscribed circle of $ABC$, and $M$ is the center of the incircle of $ABC$.

I want to express the area of the triangle $BOM$ and the radius of the incircle of $ABC$, in terms of $k$ and $\beta$.

This is what I tried: I denoted the radius of the circumscribed circle as $R$, and the radius of the incircle as $r$.
The area of $BOM$ can be obtained by: $\frac{k\cdot BD}{2}$, and $BD=\frac{BC}{2}$, I get that the area is actually: $\frac{k\cdot BC}{4}$.
Now from Law of sines I get $BC=\sin(180-2\beta)\cdot 2 R$.
So this reduces to problem to expressing $R$ (which is $OA$) in terms of $\beta$ and $k$. This is where I'm stuck. I couldn't find a way to do that.
I know that $OA=OB=R$, and that $AOB$ is also isosceles triangle, and I have a feeling it is important, but I can't figure out what am I missing...
Hints and guidelines would be very appreciated.

Assuming $BD=1$, we have $AB=AC=\frac{1}{\cos\beta}$, $[ABC]=\tan\beta$, $$R = \frac{AB}{2\sin\beta} = \frac{1}{\sin(2\beta)}$$ and: $$r = \frac{2[ABC]}{AB+AC+BC} = \frac{\tan\beta}{1+\frac{1}{\cos\beta}}=\frac{\sin\beta}{1+\cos\beta}=\tan\frac{\beta}{2}.$$ Moreover, by Euler's theorem we have: $$k^2 = OM^2 = R^2-2Rr = \frac{1}{4\sin^2\beta\cos^2\beta}-\frac{1}{\cos\beta(1+\cos\beta)}$$ so: $$k^2 = \frac{(1-2\cos\beta)^2}{4\cos^2\beta(1-\cos^2\beta)}$$ and: $$k = \frac{1-2\cos\beta}{\sin(2\beta)}.$$ This gives, in general, $$BD = \frac{\sin(2\beta)}{1-2\cos\beta} k,$$ so: $$[BOM]=\frac{k}{2}BD = \frac{\sin\beta\cos\beta}{1-2\cos\beta} k^2,\tag{1}$$ and: $$r = \frac{\sin\beta}{1+\cos\beta} BD = \frac{2(1-\cos\beta)\cos\beta}{1-2\cos\beta} k.\tag{2}$$

Hint:-

Let the equal sides be $x$,

Distance of circumcenter from A, OA = circumradius, $R=\dfrac{abc}{4\triangle}$

Distance of incenter from A, MA = $r\csc{\dfrac{A}{2}}$

You will get MA - OA in terms of $x$ and $\beta$, but MA - OA = k

Then you can substitute the value of $x$ in terms of $k$ and $\beta$ in the formula for the area, i.e.

$\dfrac{1}{2}\cdot2x\cos\beta\cdot x\sin\beta$