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AT and AT' symmetric wrt angle bisectors, 2 spheres, tetrahedron

Source: - All-Russian MO 2003 Regional (R4) 11.7

September 17, 2024
geometryangle bisector3D geometryspheretetrahedron

Problem Statement

Given a tetrahedron ABCD.ABCD. The sphere ω\omega inscribed in it touches the face ABCABC at point TT. Sphere ω′\omega' touches face ABCABC at point T′T' and extensions of faces ABDABD, BCDBCD, CADCAD. Prove that the lines ATAT and AT′AT' are symmetric wrt bisector of angle ∠BAC\angle BAC
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