3D and Limits

Geometry Level 4

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

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

Find the value of 1 k 2 L 2 \dfrac1{k^2 L^2 } .

The answer is 2.

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1 solution

Avi Solanki
Apr 24, 2017

This is a solution by my friend

@Devansh Jain .

Δ P Q R \Delta PQR is the pedal triangle of Δ A B C \Delta ABC . Incentre of the pedal triangle is the orthocentre of triangle Δ A B C \Delta ABC . I don't see why ' I I is the same for Δ P Q R \Delta PQR and Δ A B C \Delta ABC . Also,I think u need to mention that Δ A B C \Delta ABC is in the xy plane .

Sumanth R Hegde - 4 years, 1 month ago

Though the answer is correct . But I think there is a major flaw in the solution . Area of IBC is not equal to 1/3 area ABC as this triangle is not equilateral.

Ankit Kumar Jain - 2 years, 8 months ago

@Md Zuhair @Thomas Jacob @Aaron Jerry Ninan What do you guys think? Tell me if I am wrong.

Ankit Kumar Jain - 2 years, 8 months ago

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