MathDB

3

Part of 2012 Oral Moscow Geometry Olympiad

Problems(2)

line of isotomic of incircle touchpoints in equilateral is tangent to incircle

Source: 2012 Oral Moscow Geometry Olympiad grades 8-9 p3

9/8/2019
Given an equilateral triangle ABCABC and a straight line ℓ\ell, passing through its center. Intersection points of this line with sides ABAB and BCBC are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle ABCABC.
geometryisotomicmidpointincircletangent
orthocenter of the triangle DEH lies on segment CP

Source: 2012 Oral Moscow Geometry Olympiad grades 10-11 p3

9/25/2019
HH is the intersection point of the heights AA′AA' and BB′BB' of the acute-angled triangle ABCABC. A straight line, perpendicular to ABAB, intersects these heights at points DD and EE, and side ABAB at point PP. Prove that the orthocenter of the triangle DEHDEH lies on segment CPCP.
altitudeorthocenterconcurrencyconcurrentgeometry