2021年全国高中数学联合竞赛(A卷)

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第51届IMO预选题几何部分第一题为:rn题1 设锐角△ABC边BC、CA、AB的高的垂足分别为D、E、F,直线EF与△ABC的外接圆的一个交点为P,直线BP与DF交于点Q.证明:AP =AQ.[1]rn笔者在证明此题时将点C一侧补全,调整了字母顺序,得到题2.rn题2 设锐角△ABC边BC、CA、AB的高的垂足分别为D、E、F,直线EF与△ABC的外接圆的两个交点分别为M、N(M在劣弧(AB)上),直线BM与DF、CN与DE分别交于点P、Q.证明:PQ∥BC.
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几何部分rn1.已知△ABC为等腰三角形,BC=CA,D为边AB上的点,使得AD<DB,点P、Q分别为边BC、CA上的点,使得∠DPB=∠DQA=90°,PQ的中垂线与线段CQ交于点E,△ABC的外接圆与△CPQ的外接圆的第二个交点为F.若P、E、F三点共线,证明:rn∠A CB=90°.
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