Geometric Singapore Equality
Source: IMO ShortList 1988, Problem 23, Singapore 2, Problem 69 of ILL
November 9, 2005
geometrycircumcircletrigonometryincenteranalytic geometryIMO Shortlist
Problem Statement
Let be the centre of the inscribed circle of a triangle Prove that for any point
a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2,
where a \equal{} BC, b \equal{} CA and c \equal{} AB.