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tetrahedron and circumsphere

Source: 36-th Vietnamese Mathematical Olympiad 1998

February 17, 2007

geometry3D geometrytetrahedroncircumcirclegeometric transformationreflectionparallelogram

Problem Statement

Let be given a tetrahedron whose circumcenter is

O

. Draw diameters

ABCD

of the circumsphere of

ABCD

. Let

A_{0},B_{0},C_{0},D_{0}

be the centroids of triangle

BCD,CDA,DAB,ABC

. Prove that

A_{0}A_{1},B_{0}B_{1},C_{0}C_{1},D_{0}D_{1}

are concurrent at a point, say,

F

. Prove that the line through

F

and a midpoint of a side of

ABCD

is perpendicular to the opposite side.

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