MathDB

2015.11

Part of Ukraine Correspondence MO - geometry

Problems(1)

collinear wanted, 6 circumcircles related, isosceles, altitudes, midpoints

Source: 2015 All-Ukrainian Correspondence MO of magazine ''In the World of Mathematics'', grades 5-12 p11

4/27/2021
Let ABCABC be an non- isosceles triangle, HaH_a, HbH_b, and HcH_c be the feet of the altitudes drawn from the vertices A,BA, B, and CC, respectively, and MaM_a, MbM_b, and McM_c be the midpoints of the sides BCBC, CACA, and ABAB, respectively. The circumscribed circles of triangles AHbHcAH_bH_c and AMbMcAM_bM_c intersect for second time at point A′A'. The circumscribed circles of triangles BHcHaBH_cH_a and BMcMaBM_cM_a intersect for second time at point B′B'. The circumscribed circles of triangles CHaHbCH_aH_b and CMaMbCM_aM_b intersect for second time at point C′C'. Prove that points A′,B′A', B' and C′C' lie on the same line.
geometrymidpointsaltitudesisoscelesUkraine Correspondence