11.5

Part of 2005 Sharygin Geometry Olympiad

Problems(1)

existence of a point that any line passing through it bisects an unfixed angle

Source: Sharyging 2005 finals 11.5

The angle and the point

K

inside it are given on the plane. Prove that there is a point

M

with the following property: if an arbitrary line passing through intersects the sides of the angle at points

A

B

MK

is the bisector of the angle

AMB

geometryFixed pointfixedangle