MathDB

P5

Part of 2019 Dürer Math Competition (First Round)

Problems(2)

geometry , collinear points wanted

Source: 12th Dürer Math Competition ,First Round, Category E , P5

8/18/2020
Let ABCABC be a non-right-angled triangle, with AC≠BCAC\ne BC. Let FF be the midpoint of side BCBC. Let DD be a point on line ABAB satisfyingCA=CDCA=CD,and let EE be a point on line BCBC satisfying EB=EDEB = ED. The line passing through AA and parallel to EDED meets line FDFD at point II. Line AFAF meets line EDED at point JJ. Prove that points CC, II and JJ are collinear.
geometrycollinear pointsparallel
orthocenters coincide

Source: 12th Dürer Math Competition ,First Round, Category E+, P5

8/18/2020
Let ABCABC and A′B′C′A'B'C' be similar triangles with different orientation such that their orthocenters coincide. Show that lines $AA′, BB′, CC′ are concurrent or parallel.
geometrysimilar triangles