4

Part of 2007 Oral Moscow Geometry Olympiad

Problems(2)

EF tangent to circumcircle of ABC, circumcircle of BIC related, incenter I

Source: 2007 Oral Moscow Geometry Olympiad grades 8-9 p4

I

be the center of a circle inscribed in triangle

ABC

. The circle circumscribed about the triangle

BIC

intersects lines

AB

AC

E

F

, respectively. Prove that the line

EF

touches the circle inscribed in the triangle

ABC

geometrycircumcircleincentertangent

equal segments connecting opposite sides of hexagon wanted, equilateral

Source: 2007 Oral Moscow Geometry Olympiad grades 10-11 p4

The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal.
(M. Volchkevich)

geometryhexagonequal segmentsEquilateral