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prove that planes are tangent to a fixed sphere or pass through a fixed point

Source: Oral Moscow Geometry Olympiad 2015 grades 10-11 p5

August 6, 2019

geometry3D geometrysphereFixed pointplanesMoscow

Problem Statement

A triangle

ABC

and spheres are given in space

S_1

and

S_2

, each of which passes through points

A, B

and

C

. For points

M

spheres

S_1

not lying in the plane of triangle

ABC

are drawn lines

MA, MB

and

MC

, intersecting the sphere

S_2

for the second time at points

A_1,B_1

and

C_1

, respectively. Prove that the planes passing through points

A_1, B_1

and

C_1

, touch a fixed sphere or pass through a fixed point.