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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