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题 在锐角三角形ABC中,AB边上的高CE与AC边上的高BD相交于点H,以DE为直径的圆分别交AB、AC于F、G两点,FG与AH相交于点K,已知BC=25,BD=20,BE=7,求AK的长. 分析 用平面几何证法不易着手,观察图形,垂心是本图的一个特殊点,许多性质由垂心衍生出来.抓住垂心,可快速破解此题,可谓牵一发动全身.建立如图所示的直角坐标系.由已知B(7,0),C(0,24),
In the acute-angled triangle ABC, the high CE on the AB side meets the high BD on the AC side at the point H, the circle with the DE as the diameter intersects the AB, AC at the F, G points, and the FG intersects with the AH at the point K , Known BC = 25, BD = 20, BE = 7, seeking AK long. Analytical plane geometry proof method is not easy to start, observe the figure, the heart is a special point of this figure, many properties derived from the vertical center. The heart can be quickly cracked and this problem can be described as pulling a whole body. Establish a rectangular coordinate system as shown in the figure. Known by B(7,0), C(0,24),