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九○年全国高中数学联赛第二试的一道平几题是:圆O的内接四边形ABCD对角线AC、BD相交于P,△ABP、△BCP、△CDP、△DAP的外心分别为O_1、O_2、O_3、O_4,又O_1O_3与O_2O_4的交点为N,试证:P必在直线ON上。该题的一种普通解法是:(如图)∵O_1,O_4分别是△ABP、△DAP的外心,∴O_1O_2⊥AC,同理可得O_2O_3⊥AC,O_1O_2⊥BD,O_3O_4⊥BD。∴四边形O_1O_2O_3 O_4是平行四边形。
In the 90th national high school mathematics league competition in 1990, a few questions were: The inscribed quadrilateral ABCD of the circle O is diagonal AC and BD intersect at the P, and the external centers of the △ABP, △BCP, △CDP, and △DAP are The intersection of O_1, O_2, O_3, O_4, and O_1O_3 with O_2O_4 is N. For the test, P must be on the straight line ON. A common solution to this problem is: (as shown in the figure) ∵O_1, O_4 are the centers of △ABP, △DAP, respectively, ∴O_1O_2⊥AC, similarly O_2O_3⊥AC, O_1O_2⊥BD, O_3O_4⊥BD. The quadrilateral O_1O_2O_3O_4 is a parallelogram.