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在计算三角形面积公式中,常用的有:S=(1/2)ah、S=(1/2)bcsinA,从这个公式出发与三角形面积有关的性质有: 1.等底等高的两个三角形面积相等、等底(高)的两个三角形面积之比等于高(底)之比。 2.有一组内角相等(或相补)的两个三角形的面积之比等于夹这组内角的两边乘积之比。 3.相似三角形面积之比,等于相似比的平方。下面举例说明:许多与线段或角的度量关系有关的几何题,若能恰当地应用面积公式或上述有关性质,解决起来比用其它方法来得简捷明快。例1 若对角线AC将四边形ABCD分成两个相等的三角形,则AC必平分对角线BD。证明:作DE⊥AC于E,BF⊥AC于F,
In the calculation of the triangle area formula, the commonly used ones are: S=(1/2)ah, S=(1/2)bcsinA. Starting from this formula, the properties related to the area of the triangle are: 1. Two equal heights The ratio of the area of two triangles with equal area and equal bottom (height) is equal to the ratio of height (bottom). 2. The ratio of the area of two triangles with equal internal angles (or complements) is equal to the ratio of the product of the two sides of the internal angles in the group. 3. The ratio of similar triangular areas is equal to the square of the similarity ratio. The following example illustrates that many geometric problems related to the metric relationship of a line segment or an angle can be solved more simply and more quickly than other methods if the area formula or the above related properties are properly applied. Example 1 If the diagonal AC divides the quadrilateral ABCD into two equal triangles, the AC must bisect the diagonal BD. Proof: As DE⊥AC in E, BF⊥AC in F,