MathDB

G6

Part of 2008 Balkan MO Shortlist

Problems(1)

collinearity wanted, median, altitudes, 3 circles related

Source: Balkan MO Shortlist 2008 G6

4/6/2020
On triangle ABCABC the AMAM (M∈BCM\in BC) is median and BB1BB_1 and CC1CC_1 (B1∈AC,C1∈ABB_1 \in AC,C_1 \in AB) are altitudes. The stright line dd is perpendicular to AMAM at the point AA and intersect the lines BB1BB_1 and CC1CC_1 at the points EE and FF respectively. Let denoted with ω\omega the circle passing through the points E,ME, M and FF and with ω1\omega_1 and with ω2\omega_2 the circles that are tangent to segment EFEF and with ω\omega at the arc EFEF which is not contain the point MM. If the points PP and QQ are intersections points for ω1\omega_1 and ω2\omega_2 then prove that the points P,QP, Q and MM are collinear.
geometrycollinearcirclestangentaltitudesmedian