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三角形面积公式,不仅可用来计算有关图形的面积,而且在证题方面也有较广泛的应用。本文仅就用它来证明有关成比例线段略举几例,思路常是运用面积相等或面积之比使其获证。若恰当地运用三角函数关系往往更为简便。例1 圆内接四边形ABCD的对角线AC平分另一对角线BD于E,求证:AB/AD=DC/BC。分析:结论即求证:AB·BC=AD·DC,∠ABC=180°-∠ADC,于是变为求证: (1/2)AB·BCsin∠ABC=(1/2)AD·DCsin(180°-∠ADC), 根据三角形面积公式,可考虑S_(△ABC)=S_(△ADC)。
The triangular area formula can be used not only to calculate the area of the relevant figure, but also has a wider range of applications in the subject matter. This article only uses it to prove that there are several examples of proportional line segments. The idea is often to use the ratio of equal area or area to make it known. It is often easier to use trigonometric functions properly. Example 1 The diagonal AC of the circle inscribed quadrangle ABCD bisects the other diagonal BD at E, verifying: AB/AD=DC/BC. Analysis: The conclusion is to verify: AB·BC=AD·DC, ∠ABC=180°-∠ ADC, then change to verification: (1/2)AB·BCsin∠ABC=(1/2)AD·DCsin(180° - ∠ ADC) According to the triangle area formula, S_(ΔABC) = S_(ΔADC) can be considered.