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单词 ProofOfPivotTheorem
释义

proof of pivot theorem


Let ABC be a triangle, and let D, E, and F be pointson BC, CA, and AB, respectively. The circumcirclesMathworldPlanetmath ofAEF and BFD intersect in F and in anotherpoint, which we call P. Then AEPF and BFPD are cyclicquadrilateralsMathworldPlanetmath, so

A+EPF=π

and

B+FPD=π

Combining this with A+B+C=π and EPF+FPD+DPE=2π, we get

C+DPE=π.

This implies that CDPE is a cyclic quadrilateral as well, so thatP lies on the circumcircle of CDE. Therefore, thecircumcircles of the triangles AEF, BFD, and CDE have a commonpoint, P.

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更新时间:2025/5/4 8:22:29