MathDB

3

Part of 2018 South East Mathematical Olympiad

Problems(2)

Prove line tangent to circle given segment is bisected

Source: CSEMO 2018 Grade 10 Q3

7/30/2018
Let OO be the circumcenter of acute △ABC\triangle ABC(AB<ACAB<AC), the angle bisector of ∠BAC\angle BAC meets BCBC at TT and MM is the midpoint of ATAT. Point PP lies inside △ABC\triangle ABC such that PB⊥PCPB\perp PC. D,ED,E distinct from PP lies on the perpendicular to APAP through PP such that BD=BP,CE=CPBD=BP, CE=CP. If AOAO bisects segment DEDE, prove that AOAO is tangent to the circumcircle of △AMP\triangle AMP.
geometry
Prove angle relation given parallelogram

Source: CSEMO 2018 Grade 11 Q3

7/30/2018
Let OO be the circumcenter of △ABC\triangle ABC, where ∠ABC>90∘\angle ABC> 90^{\circ} and MM is the midpoint of BCBC. Point PP lies inside △ABC\triangle ABC such that PB⊥PCPB\perp PC. D,ED,E distinct from PP lies on the perpendicular to APAP through PP such that BD=BP,CE=CPBD=BP, CE=CP. If quadrilateral ADOEADOE is a parallelogram, prove that ∠OPE=∠AMB.\angle OPE = \angle AMB.
geometryparallelogram