# 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.

###### It is a challenge for all those who love geometry! This is the toughest problem that I have posted on Brilliant till now.

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