It's staring at me

Geometry Level 5

Let \(\Delta ABC\) be a triangle with vertices \( A(0,0),B(6,0), C(3,3\sqrt{3})\), and \(\Delta PQR\) the pedal triangle of \(\Delta ABC\).

The sum of the circumradius of \( \Delta ABC\) and inradius of \( \Delta PQR\) can be expressed as \(\frac{a\sqrt{b}}{c}\) for positive integers \(a,b,\) and \(c,\) where \(a,c\) are coprime and \(b\) is square-free.

Find \(a\times b\times c\).

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Aniket Verma by Aniket Verma
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