MathDB

g1

Part of Indonesia MO Shortlist - geometry

Problems(2)

collinear wanted, angle bisector, 2 parallel, 1 perpendicular

Source: Indonesia INAMO Shortlist 2009 G1 https://artofproblemsolving.com/community/c1101409_

12/10/2021
Given triangle ABCABC, ALAL bisects angle ∠BAC\angle BAC with LL on side BCBC. Lines LRLR and LSLS are parallel to BABA and CACA respectively, RR on side ACAC andS S on side ABAB, respectively. Through point BB draw a perpendicular on ALAL, intersecting LRLR at MM. If point DD is the midpoint of BCBC, prove that that the three points A,M,DA, M, D lie on a straight line.
collineargeometryangle bisector
DE, DF tangents to circle (AEF)

Source: Indonesia INAMO Shortlist 2010 G1

8/27/2021
In triangle ABCABC, let DD be the midpoint of BCBC, and BEBE, CFCF are the altitudes. Prove that DEDE and DFDF are both tangents to the circumcircle of triangle AEFAEF
geometrytangentalttiudes