MathDB

G4

Part of 2020 Balkan MO Shortlist

Problems(1)

OK _|_ TM wanted, starting with isosceles trapezoid

Source: 2020 Balkan MO shortlist G4

9/14/2021
Let MAZNMAZN be an isosceles trapezium inscribed in a circle (c)(c) with centre OO. Assume that MNMN is a diameter of (c)(c) and let B B be the midpoint of AZAZ. Let (ϵ)(\epsilon) be the perpendicular line on AZAZ passing through A A. Let CC be a point on (ϵ)(\epsilon), let EE be the point of intersection of CBCB with (c)(c) and assume that AEAE is perpendicular to CBCB. Let DD be the point of intersection of CZCZ with (c)(c) and let FF be the antidiametric point of DD on (c)(c). Let P P be the point of intersection of FEFE and CZCZ. Assume that the tangents of (c)(c) at the points MM and ZZ meet the lines AZAZ and PAPA at the points KK and TT respectively. Prove that OKOK is perpendicular to TMTM.
Theoklitos Parayiou, Cyprus
perpendiculartrapezoidgeometry