# 3D and Limits

Geometry Level 4

The incenter $$I$$ of the triangle $$PQR$$ is the foot of the normal from the point $$M = (1,2,6)$$ to the $$xy$$-plane, where $$P,Q,R$$ are the feet of altitudes of an isosceles triangle $$ABC$$ whose vertex is $$A$$ and base $$BC$$ of 6 unit length.

Let $$\displaystyle \lim_{A\to {\frac \pi 2}^+} \dfrac{e^v - e^k}{\sqrt{1- \sin A}} = \dfrac{e^k}L$$ for integer $$k$$, where $$v$$ is the volume of the tetrahedron $$MIBC$$.

Find the value of $$\dfrac1{k^2 L^2 }$$.

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