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单词 ProofOfPtolemysTheorem
释义

proof of Ptolemy’s theorem


Let ABCD be a cyclic quadrialteral. We will prove that

ACBD=ABCD+BCDA.

Find a point E on BD such that BCA=ECD. Since BAC=BDC for opening the same arc, we have triangle similarityMathworldPlanetmathABCDEC and so

ABDE=CACD

which implies ACED=ABCD.

Also notice that ADCBEC since have two pairs of equal angles. The similarity implies

ACBC=ADBE

which implies ACBE=BCDA.

So we finally have ACBD=AC(BE+ED)=ABCD+BCDA.

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更新时间:2025/5/4 6:54:36