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[几何] 内心有关证明题

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星奔川骛 posted 2025-4-29 22:52 |Read mode
Last edited by hbghlyj 2025-5-1 17:23三角形ABC的内心I,作ID垂直BC,ID交圆(BIC)于E,F,G是圆(BIC)上的点满足AFG共线,J为BC与EG的交点,证明:IJ⊥EF

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乌贼 posted 2025-6-21 11:12
如图: 1.png
    $ O $为圆心,$ M,H $关于$ BC $对称。有\[ \begin{cases}
\triangle AIC\sim \triangle CHP\\\triangle MQC\sim \triangle ICF
\end{cases} \]得\[ \triangle QHP\sim \triangle AIF\riff \angle QPH=\angle IFG=\angle IEJ\riff JQ\px EP \]即有$ JQPE $和$ JQMI $都为平行四边形,所以\[ IJ\px MF\riff IJ\perp EF \]

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