# A geometry problem by Akshay Yadav

Geometry Level 5

The figure represents above shows a Cartesian plane.

In the given figure $$O= (0,0)$$ is the center of a circle with radius $$\dfrac{5}{2}$$ units. $$\angle ABC =18^\circ$$ and $$\angle ADC = 9^\circ$$.

If the $$x$$-coordinates of $$D$$ are in form

$\dfrac{a(\sqrt{a}-b)(1+\sqrt{a}-\sqrt{ac+c\sqrt{a}})}{c^2(c-c\sqrt{a}+\sqrt{\frac{c}{a+\sqrt{a}}}+\sqrt{\frac{ac}{a+\sqrt{a}}}-c\sqrt{ac+c\sqrt{a}}+\sqrt{a}\sqrt{ac+c\sqrt{a}})}$

Note that $$a$$, $$b$$ and $$c$$ are mutually prime positive integers.

Find $$a+b+c$$ .

You may use the fact that $$\sin 18^\circ=\dfrac{\sqrt{5}-1}{4}$$.

Clarification: Angles are measured in degrees.

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