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2022IMO几何题,我也会做?

2023-04-29 09:21 作者:奥博格沙特  | 我要投稿

2022IMO Problem 1.4.

ABCDE为一凸五边形满足BC%3DDE%2C%5C%20假设在ABCDE内部存在一点T使得TB%3DTD%2C%5C%20TC%3DTE%E2%88%A0ABT%3D%E2%88%A0TEA.%5C%20令直线AB分别与直线CDCT交于点PQ,假设P%2CB%2CA%2CQ在同一直线上按照此顺序排列。令直线AE分别与直线CDDT交于点RS,假设R%2CE%2CA%2CS在同一直线上按照此顺序排列。证明P%2CS%2CQ%2CR落在同一个圆上。

证明

由条件易知%E2%96%B3TBC%5C%20%E2%89%8C%5C%20%E2%96%B3TDE

%5Cimplies%20%E2%88%A0BTC%3D%E2%88%A0ETD

%5Cimplies%20%E2%88%A0BTQ%3D%E2%88%A0ETS

%E2%88%A0QBT%3D%E2%88%A0SET

%5Cimplies%20%E2%96%B3BTQ%E2%88%BD%E2%96%B3ETS

%5Cimplies%20%E2%88%A0BQT%3D%E2%88%A0EST

        %5Cfrac%7BTS%7D%7BTQ%7D%3D%5Cfrac%7BTE%7D%7BTB%7D%3D%5Cfrac%7BTC%7D%7BTD%7D

%5Cimplies%20TS%20%5Ccdot%20TD%20%3D%20TQ%20%5Ccdot%20TC

%5Cimplies%20C%2C%20D%2C%20Q%2C%20S共圆

%5Cimplies%20%E2%88%A0SQC%3D%E2%88%A0SDC

%5Cimplies%20%E2%88%A0SQC-%E2%88%A0BQT%3D%E2%88%A0SDC-%E2%88%A0EST

%5Cimplies%20%E2%88%A0PQS%3D%E2%88%A0PRS

%5Cimplies%20P%2C%20S%2C%20Q%2C%20R共圆

Q.E.D.

说明  本题的思路是从条件出发,寻找图形的性质,进而得出结论。


本文中的方法仅为个人方法,如有雷同,纯属巧合.

如果读者有其他方法,或者有问题,欢迎分享交流!

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