Let Q be the centre of the inscribed circle of a triangle ABC. Prove that for any point P,
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. geometrycircumcircletrigonometryincenteranalytic geometryIMO Shortlist