MathDB

2

Part of 2023 Puerto Rico Team Selection Test

Problems(2)

isosceles wanted, semicircle related Puerto Rico TST 2023.1.2

Source:

3/23/2024
Consider a semicircle with center MM and diameter ABAB. Let PP be a point in the semicircle, different from AA and BB, and let QQ be the midpoint of the arc APAP. The line parallel to QPQP through MM intersects PBPB at the point SS. Prove that the triangle PMSPMS is isosceles.
geometryisoscelessemicircle
AP _|_ BP wanted, 2 touchpoints of incircle related

Source: Puerto Rico TST 2023.2.2

3/24/2024
Let II be the incenter of a triangle ABCABC and let DD and EE be the touchpoints of the incircle with sides BCBC and ACAC, respectively. The lines DEDE and BIBI intersect at point PP. Prove that APAP is perpendicular to BPBP.
geometryperpendicularperpendicularity