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Part of 2013 Oral Moscow Geometry Olympiad

Problems(2)

midpoint of KI lies on AC, angle bisectors related, right triangle

Source: 2013 Oral Moscow Geometry Olympiad grades 8-9 p3

8/14/2019
The bisectors AA1AA_1 and CC1CC_1 of the right triangle ABCABC (∠B=90o\angle B = 90^o) intersect at point II. The line passing through the point C1C_1 and perpendicular on the line AA1AA_1 intersects the line that passes through A1A_1 and is perpendicular on CC1CC_1, at the point KK. Prove that the midpoint of the segment KIKI lies on segment ACAC.
geometryangle bisectorright trianglemidpoint
is there polyhedron whose area ratio of any two faces is at least 2

Source: 2013 Oral Moscow Geometry Olympiad grades 10-11 p3

8/16/2019
Is there a polyhedron whose area ratio of any two faces is at least 22 ?
polyhedronarearatio3D geometrygeometry