4

Part of 2006 Oral Moscow Geometry Olympiad

Problems(2)

construct line passing through B so that inradii of 2 small triangles are equal

Source: 2006 Oral Moscow Geometry Olympiad grades 8-9 p4

An arbitrary triangle

ABC

is given. Construct a straight line passing through vertex

B

and dividing it into two triangles, the radii of the inscribed circles of which are equal.
(M. Volchkevich)

geometryequal circlesinradiusconstruction

equal products of distances,cyclic ABCD, tangents concurrent with symmetric line

Source: 2006 Oral Moscow Geometry Olympiad grades 10-11 p4

ABCD

is inscribed in a circle, the center

O

of which lies inside it. The tangents to the circle at points

A

C

and a straight line, symmetric to

BD

O

, intersect at one point. Prove that the products of the distances from

O

to opposite sides of the quadrilateral are equal.
(A. Zaslavsky)

geometrySymmetricdistancecyclic quadrilateral