# Chilling Geometry

Geometry Level 5

Let $O$ be the circumcentre of an acute triangle $\Delta ABC$, with sides equal to $5$, $6$, and $7$ units in length, and with orthocentre $H$. Let $P$ be any point on the major arc $BC$ (the arc not containing point $A$) of the circumcircle of $\Delta ABC$ , except $B$ and $C$. Let $D$ be a point outside $\Delta ABC$ such that $\overline{AD}=\overline{PC}$ and $AD || PC$. Let $K$ be the orthocentre of $\Delta ACD$. The distance of the circumcentre of $\Delta ABC$ to $K$ can be expressed as $\dfrac{a\sqrt{b}}{4c}$, where $a,b,c \in \mathbb{Z^+}$ ,$\gcd(a,c)=1$ and $b$ does not contain the square of any prime number in its prime factorization. Determine the value of $a+b+c$.

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