MathDB

3

Part of 2013 Benelux

Problems(1)

Triple product of three sides equals another triple product

Source: Benelux MO 2013 Q3

4/29/2013
Let △ABC\triangle ABC be a triangle with circumcircle Γ\Gamma, and let II be the center of the incircle of △ABC\triangle ABC. The lines AIAI, BIBI and CICI intersect Γ\Gamma in D≠AD \ne A, E≠BE \ne B and F≠CF \ne C. The tangent lines to Γ\Gamma in FF, DD and EE intersect the lines AIAI, BIBI and CICI in RR, SS and TT, respectively. Prove that ∣AR∣⋅∣BS∣⋅∣CT∣=∣ID∣⋅∣IE∣⋅∣IF∣.\vert AR\vert \cdot \vert BS\vert \cdot \vert CT\vert = \vert ID\vert \cdot \vert IE\vert \cdot \vert IF\vert.
geometrycircumcircleratiopower of a pointgeometry unsolved