Chilling Geometry

Geometry Level 5

Let OO be the circumcentre of an acute triangle ΔABC\Delta ABC, with sides equal to 55, 66, and 77 units in length, and with orthocentre HH. Let PP be any point on the major arc BCBC (the arc not containing point AA) of the circumcircle of ΔABC\Delta ABC , except BB and CC. Let DD be a point outside ΔABC\Delta ABC such that AD=PC\overline{AD}=\overline{PC} and ADPCAD || PC. Let KK be the orthocentre of ΔACD\Delta ACD. The distance of the circumcentre of ΔABC\Delta ABC to KK can be expressed as ab4c\dfrac{a\sqrt{b}}{4c}, where a,b,cZ+a,b,c \in \mathbb{Z^+} ,gcd(a,c)=1\gcd(a,c)=1 and bb does not contain the square of any prime number in its prime factorization. Determine the value of a+b+ca+b+c.

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