MathDB

1

Part of 2024 Malaysian IMO Training Camp

Problems(2)

Cyclic implies parallel?

Source: Own. Malaysian IMO TST 2024 P1

4/21/2024
Let ABCABC be an acute triangle with orthocenter HH, and let BEBE and CFCF be the altitudes of the triangle. Choose two points PP and QQ on rays BHBH and CHCH respectively, such that:
∙\bullet PQPQ is parallel to BCBC;
∙\bullet The quadrilateral APHQAPHQ is cyclic.
Suppose the circumcircles of triangles APFAPF and AQEAQE meet again at X≠AX\neq A. Prove that AXAX is parallel to BCBC.
Proposed by Ivan Chan Kai Chin
geometry
Proving lengths are equal

Source: Malaysian SST 2024 P1

9/5/2024
A cyclic quadrilateral ABCDABCD has diameter ACAC with circumcircle ω\omega. Let KK be the foot of the perpendicular from CC to BDBD, and the tangent to ω\omega at AA meets BDBD at TT. Let the line AKAK meets ω\omega at XX and choose a point YY on line AKAK such that ∠TYA=90∘\angle TYA=90^{\circ}. Prove that AY=KXAY=KX.
Proposed by Anzo Teh Zhao Yang
geometry