In triangle ABC, AL, BM, and CN are concurrent at point P. Points U,.pdf

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In triangle ABC, AL, BM, and CN are concurrent at point P. Points U, V, and W are chosen on AB, AC, and BC, respectively, so that LU is parallel to AC, NV is parallel to BC, and MW is parallel to AB. Prove that AW, BV, and CU are concurrent (at point K) Solution From Ceva Theorem, since AL,BM,CN are concurrent, we have BL/CL*CM/AM*AN/BN=1 LU//AC implies BL/CL=BU/AU NV/BC implies AN/BN=AV/CV MW//AB implies CM/AM=CW/BW So from all four above equations we get BU/AU*CW/BW*AV/CV=1 which again by Ceva Theorem we get AW,BV,CU are concurrent.

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