MathDB

2

Part of 2004 IberoAmerican

Problems(2)

Find the locus of point

Source: IberoAmerican Mathematical Olympiad 2004, Problem 2

5/23/2007
In the plane are given a circle with center O O and radius r r and a point A A outside the circle. For any point M M on the circle, let N N be the diametrically opposite point. Find the locus of the circumcenter of triangle AMN AMN when M M describes the circle.
geometrycircumcircleinductionratiogeometric transformationreflectionpower of a point
Collinear

Source: IberoAmerican Mathematical Olympiad 2004, Problem 5

9/23/2004
Given a scalene triangle ABC ABC. Let A′ A', B′ B', C′ C' be the points where the internal bisectors of the angles CAB CAB, ABC ABC, BCA BCA meet the sides BC BC, CA CA, AB AB, respectively. Let the line BC BC meet the perpendicular bisector of AA′ AA' at A′′ A''. Let the line CA CA meet the perpendicular bisector of BB′ BB' at B′ B'. Let the line AB AB meet the perpendicular bisector of CC′ CC' at C′′ C''. Prove that A′′ A'', B′′ B'' and C′′ C'' are collinear.
geometrycircumcircleperpendicular bisector