MathDB

Geometric inequality

Source: Bulgarian IMO TST 2008, Day 1, Problem 2

July 8, 2013

inequalitiestrigonometrycomplex numbersinequalities proposedgeometry

Problem Statement

The point

P

lies inside, or on the boundary of, the triangle

ABC

. Denote by

d_{a}

d_{b}

and

d_{c}

the distances between

P

and

BC

CA

, and

AB

, respectively. Prove that

\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}

. When does the equality holds?