MathDB

5

Part of 2023 Centroamerican and Caribbean Math Olympiad

Problems(1)

Two points lying on a triangle's circumcircle

Source: 2023 Centroamerican and Caribbean Math Olympiad, P5

7/26/2023
Let ABCABC be an acute-angled triangle with AB<ACAB < AC and Γ\Gamma the circumference that passes through A, BA,\ B and CC. Let DD be the point diametrically opposite AA on Γ\Gamma and ℓ\ell the tangent through DD to Γ\Gamma. Let P,QP, Q and RR be the intersection points of BCB C with ℓ\ell, of APA P with Γ\Gamma such that Q≠AQ \neq A and of QDQ D with the AA-altitude of the triangle ABCABC, respectively. Define SS to be the intersection of ABAB with ℓ\ell and TT to be the intersection of ACA C with ℓ\ell. Show that SS and TT lie on the circumference that passes through A,QA, Q and RR.
geometrycircumcircle