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3 midpoints of edges in tetrahedron and tangent's intersection, coplanar

Source: All-Russian MO 2023 Final stage 11.6

August 26, 2024

geometry3D geometrytetrahedron

Problem Statement

The plane

\alpha

intersects the edges

AB

,

BC

,

T

and

DA

of the tetrahedron

ABCD

at points

X, Y, Z

and

P T

, respectively. It turned out, that points

Y

and

T

lie on a circle

\omega

constructed with segment

XZ

as the diameter. Point

P

is marked in the plane

\alpha

so that the lines

P Y

and

P T

are tangent to the circle

\omega

.Prove that the midpoints of the edges are

AB

,

BC

,

CD,

DA

and the point

P

lie in the same plane.

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