MathDB

3

Part of 2017 Danube Mathematical Olympiad

Problems(2)

Perpendicular Contains Midpoint

Source: Mathematical Danube Competition 2017, Juniors P3

4/21/2022
Consider an acute triangle ABCABC in which A1,B1,A_1, B_1, and C1C_1 are the feet of the altitudes from A,B,A, B, and C,C, respectively, and HH is the orthocenter. The perpendiculars from HH onto A1C1A_1C_1 and A1B1A_1B_1 intersect lines ABAB and ACAC at PP and Q,Q, respectively. Prove that the line perpendicular to B1C1B_1C_1 that passes through AA also contains the midpoint of the line segment PQPQ.
geometryromania
Problem 3

Source: Danube Mathematical Competition 2017, Romania

10/28/2017
Let O,HO,H be the circumcenter and the orthocenter of triangle ABCABC. Let FF be the foot of the perpendicular from C onto AB, and MM the midpoint of CHCH. Let N be the foot of the perpendicular from C onto the parallel through H at OMOM. Let DD be on ABAB such that CA=CDCA=CD. Let BNBN intersect CDCD at PP. Let PHPH intersect CACA at QQ. Prove that QF⊥OFQF\perp OF.
geometrycircumcircle