# CD perpendicular to AB

Geometry Level 4

Let $$ABC$$ be a triangle right angled at $$C$$. We are given that $$CD \perp AB$$ and $$F$$ is the center of the inscribed circles of the triangles $$ADC$$ and $$BDC$$. Parallel lines through $$E$$ and $$F$$ with $$CD$$ meet $$AC$$ and $$BC$$ at points $$E'$$ and $$F'$$.

Denote $$n = \frac{CE'}{CF'}$$ and that $$n$$ is a rational number of the form $$\frac pq$$ for coprime positive integers $$p$$ and $$q$$. Find the value of $$p^3 + q^3$$.

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