Which point?

Geometry Level 3

A point P P is located in triangle A B C . ABC. Let X , Y , Z X, Y, Z be the feet of the perpendiculars dropped from P P to sides B C , C A , A B , BC, CA, AB, respectively. If P X + P Y + P Z = 2017 , PX + PY + PZ = 2017, find the minimum possible value of P A + P B + P C . PA + PB + PC.


The answer is 4034.

Steven Yuan by Steven Yuan , 20, USA

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1 solution

Steven Yuan
Jun 16, 2017

By the Erdös-Mordell Inequality ,

P A + P B + P C ≥ 2 ( P X + P Y + P Z ) . PA + PB + PC \geq 2(PX + PY + PZ).

Thus, P A + P B + P C ≥ 2 ( 2017 ) = 4034 . PA + PB + PC \geq 2(2017) = \boxed{4034}. Equality is achieved when A B C ABC is equilateral and P P is its circumcenter/orthocenter/incenter/centroid.

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