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单词 ProofOfVanAubelTheorem
释义

proof of Van Aubel theorem


We want to prove

CPPF=CDDB+CEEA

On the picture, let us call ϕ to the angle ABE and ψ to the angle EBC.

A generalizationPlanetmathPlanetmath of bisectorMathworldPlanetmath’s theorem states

CEEA=CBsinψABsinϕon ABC

and

CPPF=CBsinψFBsinϕon FBC.

From the two equalities we can get

CEABEA=CPFBPF

and thus

CPPF=CEABEAFB.

Since AB=AF+FB, substituting leads to

CEABEAFB=CE(AF+FB)EAFB
=CEAFEAFB+CEFBEAFB
=CEAFEAFB+CEEA

But Ceva’s theorem states

CEEAAFFBBDDC=1

and so

CEAFEAFB=CDDB

Subsituting the last equality gives the desired result.

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更新时间:2025/5/4 22:33:09