Let ABC be a triangle with AB=13, BC=14, and AC=15. Let M be the midpoint of BC and let Γ be the circle passing through A and tangent to line BC at M. Let Γ intersect lines AB and AC at points D and E, respectively, and let N be the midpoint of DE. Suppose line MN intersects lines AB and AC at points P and O, respectively. If the ratio MN:NO:OP can be written in the form a:b:c with a,b,c positive integers satisfying gcd(a,b,c)=1, find a+b+c.James Tao Online Math Openratiotrigonometrygeometrysimilar trianglestrig identitiesLaw of Cosines