5

Part of 2012 Oral Moscow Geometry Olympiad

Problems(2)

fixed point, related to circumcircles and orthocenter

Source: 2012 Oral Moscow Geometry Olympiad grades 8-9 p5

Given a circle and a chord

AB

, different from the diameter. Point

C

moves along the large arc

AB

. The circle passing through passing through points

A, C

H

of intersection of altitudes of of the triangle

ABC

, re-intersects the line

BC

P

. Prove that line

PH

passes through a fixed point independent of the position of point

C

geometrycircumcircleorthocenterfixedFixed point

min sum of distances of a point of circle from equidistant points from center

Source: 2012 Oral Moscow Geometry Olympiad grades 10-11 p5

Inside the circle with center

O

A

B

are marked so that

OA = OB

M

on the circle from which the sum of the distances to points

A

B

is the smallest among all possible.

minimumSumdistancegeometrycircle