# Mystic cevians

Geometry Level 5

Let $$H$$ be the orthocentre and $$O$$ the circumcentre of a triangle $$ABC$$ with the length of side $$BC$$ equal to $$2014$$. Cevians are constructed from vertices $$B$$ and $$C$$ so as to be tangent to $$\gamma$$ , the circle passing through $$A$$ and $$H$$ and congruent to the circle centred at $$O$$ touching $$BC$$.

Let the cevians through $$B$$ and $$C$$ touch the circle $$\gamma$$ at $$T$$ and $$U$$ respectively. If $$BT$$ and $$CU$$ are of integer lengths, what is their sum?

Aside: On the query of some, this is indeed an original problem of mine!

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