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单词 ProofOfSimsonsLine
释义

proof of Simson’s line


Given a ABC with a point P on its circumcircleMathworldPlanetmath (other than A,B,C),we will prove that the feet of the perpendicularsMathworldPlanetmathPlanetmathPlanetmathPlanetmath drawn from P to the sides AB,BC,CA(or their prolongations) are collinearMathworldPlanetmath.

Since PW is perpendicular to BW and PU is perpendicular to BUthe point P lies on the circumcircle of BUW.

By similarMathworldPlanetmathPlanetmath arguments, P also lies on the circumcircle of AWV and CUV.

This implies that PUBW , PUCV and PVWA are all cyclic quadrilateralsMathworldPlanetmath.

Since PUBW is a cyclic quadrilateral,

UPW=180-UBW

implies

UPW=180-CBA

Also CPAB is a cyclic quadrilateral, therefore

CPA=180-CBA

(opposite angles in a cyclic quarilateral are supplementaryPlanetmathPlanetmath).

From these two, we get

UPW=CPA

Subracting CPW, we have

UPC=WPA

Now, since PVWA is a cyclic quadrilateral,

WPA=WVA

also, since UPVC is a cyclic quadrilateral,

UPC=UVC

Combining these two results with the previous one, we have

WVA=UVC

This implies that the points U,V,W are collinear.

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更新时间:2025/5/4 4:49:28