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面积比的类型很多,本文着重谈“有一个角对应相等(或互补)的两个三角形面积之比等于夹这个角的两边乘积之比”在几何证题中的广泛应用。这个性质可表示为: 定理:在△ABC与△A_1B_1C_1中,∠B=∠B_1(或互补),则 S_(△ABC)/S(△A_1B_1C_1)=(AB·BC)/(A_1B_1·B_1C_1)。我们用三角形的面积公式S=1/2acsinB容易证明上述定理(略)。不少比例线段的证明,可归结为这个性质的应用。下面举例说明之。 1.证明三角形内角平分线的性质例1 已知△ABC的内角A的平分线交BC于D 求证:
There are many types of area ratios. This article focuses on the wide application of “the ratio of the ratio of two triangles with a corner corresponding to equal (or complementary) is equal to the ratio of two sides with this angle”. This property can be expressed as: Theorem: In ΔABC and △A_1B_1C_1, ∠B=∠B_1 (or complement), then S_(ΔABC)/S(△A_1B_1C_1)=(AB·BC)/(A_1B_1·B_1C_1) . We can easily prove the above theorem by using the area formula of the triangle S=1/2acsinB (omitted). Many proofs of the proportional line segment can be attributed to the application of this nature. Here’s an example. 1. Proof of the property of the bisector of the inner angle of the triangle Example 1 It is known that the bisector of the inner angle A of the △ABC is intersected by BC in the verification of D: