# First take reflection, then cyclic

Geometry Level 5

In figure below,

$$ABCD$$ is a convex quadrilateral with $$\angle DAB = \angle BDC = { 90 }^{ o }$$. Let the incircles of triangles $$ABD$$ and $$BCD$$ touch $$BD$$ at $$P$$ and $$Q$$. If $$AD = 999$$ and $$PQ = 200$$, find the sum of the inradii of two in-circles.

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