MathDB

6

Part of 2005 Oral Moscow Geometry Olympiad

Problems(2)

perpendicular wanted, incenter, midpoints related

Source: 2005 Oral Moscow Geometry Olympiad grades 8-9 p6

10/16/2020
Let A1,B1,C1A_1,B_1,C_1 are the midpoints of the sides of the triangle ABC,IABC, I is the center of the circle inscribed in it. Let C2C_2 be the intersection point of lines C1IC_1 I and A1B1A_1B_1. Let C3C_3 be the intersection point of lines CC2CC_2 and ABAB. Prove that line IC3IC_3 is perpendicular to line ABAB.
(A. Zaslavsky)
geometryincenterperpendicularmidpoints
for any 3 of 6 lines, exists a 4th such that they are tangent to a circle

Source: 2005 Oral Moscow Geometry Olympiad grades 10-11 p6

10/21/2020
Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle?
(I. Bogdanov)
combinatorial geometrygeometryTangents