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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