TopsOJ Summer 2025 - Individual Round - Problem 15

Let \triangle ABC have AB=10, BC=14, and CA=8\sqrt{2}. Let H and O be the orthocenter and circumcenter, respectively. Let AO intersect the line at C perpendicular to BC at D, and let the reflection of D across BC be D'. Define E on (ABH) such that \angle DAD'=\angle HAE. OE can be expressed as \frac{a\sqrt{b}}{c} where a, b, and c are positive integers, \gcd{(a, c)} = 1, and b is not divisible by the square of any prime. Compute a+b+c.