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单词 ProofOfMenelausTheorem
释义

proof of Menelaus’ theorem


First we note that there are two different cases: Either the line connecting X, Y and Z intersects two sides of the triangleMathworldPlanetmath or none of them. So in the first case that it intersects two of the triangle’s sides we get the following picture:

From this we follow (h1, h2 and h3 being undircted):

AZZB=-h1h2
BYYC=h2h3
CXXA=h3h1.

Mulitplying all this we get:

AZZBBYYCCXXA=-h1h2h3h2h3h1=-1.

The second case is that the line connecting X, Y and Z does not intersect any of the triangle’s sides:

In this case we get:

AZZB=-h1h2
BYYC=-h2h3
CXXA=-h3h1.

So multiplicationPlanetmathPlanetmath again yields Menelaus’ theoremMathworldPlanetmath.

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更新时间:2025/5/4 16:54:29