MathDB

2

Part of 2012 Irish Math Olympiad

Problems(2)

Tangent Circumcircles

Source: 2012 Paper 1 Problem 2

2/16/2018
A,B,CA,B,C and DD are four points in that order on the circumference of a circle KK. ABAB is perpendicular to BCBC and BCBC is perpendicular to CDCD. XX is a point on the circumference of the circle between AA and DD. AXAX extended meets CDCD extended at EE and DXDX extended meets BABA extended at FF. Prove that the circumcircle of triangle AXFAXF is tangent to the circumcircle of triangle DXEDXE and that the common tangent line passes through the center of the circle KK.
geometrycircumcircle
Angle Bisector Intersects Circumcircle

Source: 2012 IrMO Paper 2 Problem 2

2/16/2018
Consider a triangle ABCABC with ∣AB∣≠∣AC∣|AB|\neq |AC|. The angle bisector of the angle CABCAB intersects the circumcircle of △ABC\triangle ABC at two points AA and DD. The circle of center DD and radius ∣DC∣|DC| intersects the line ACAC at two points CC and B’B’. The line BB’BB’ intersects the circumcircle of △ABC\triangle ABC at BB and EE. Prove that B’B’ is the orthocenter of △AED\triangle AED.
geometryangle bisectorcircumcircle