# JOMO 5, Short 10

Geometry Level 5

In the Diagram, $$AB=10, BC=24$$. $$I$$ is the Incenter of $$\triangle ABC$$, $$G$$ is the Centroid of $$\triangle ABC$$. $$O$$ and $$E$$ are the midpoints of $$AC$$ and $$BC$$ respectively.

The product of $$OG, BG, BI, IE, OE, GE, GK, KE$$ and $$IK$$ can be expressed as $$\frac{a\sqrt b}c$$ where $$a,b,c\in \mathbb Z^+$$, $$gcd(a,c)=1$$, and $$b$$ is not divisible by the square of a integer larger than 1. Find the last 3 digits of $$a+b+c$$

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