# Tor-tiled Triangles

Geometry Level 4

Equilateral triangle $$A_1B_1C_1$$ has side length $$1$$. A median is drawn from point $$A_1$$ to hit $$B_1C_1$$ at $$A_2$$. $$B_1$$ is renamed $$C_2$$ and $$A_1$$ is renamed $$B_2$$. Now, the same maneuver is done to triangle $$A_2B_2C_2$$ to produce triangle $$A_3B_3C_3$$, and so on, until infinity. Points $$A_n,B_n,C_n$$ ultimately coincide to point $$P$$ as $$n$$ approaches infinity.$AP^2+BP^2+CP^2$ can be expressed as $$\dfrac{p}{q}$$ for positive coprime integers $$p,q$$. Find $$p+q$$.

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