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bisectors of plane <KMQ and <LMP are the same, triangular pyramid, spheres

Source: All-Russian MO 2004 Regional (R4) 11.8

September 27, 2024
geometry3D geometryAngle Bisectorsangle bisectorpyramidSpheressphere

Problem Statement

Given a triangular pyramid ABCDABCD. Sphere S1S_1 passing through points AA, BB, CC, intersects edges ADAD, BDBD, CDCD at points KK, LL, MM, respectively; sphere S2S_2 passing through points AA, BB, DD intersects the edges ACAC, BCBC, DCDC at points PP, QQ, MM respectively. It turned out that KL∥PQKL \parallel PQ. Prove that the bisectors of plane angles KMQKMQ and LMPLMP are the same.
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