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根据某些条件若能构造出直角三角形,利用直角三角形的性质,可使某些无理不等式得到直观巧妙的证法,下举几例以说明之: 例1 设a≥c,b≥c,c≥0,求证并确定等号成立的条件。证明:如图1,在长度为c~(1/2)的线段BC上作Rt△ABE和Rt△ECD,使AB=b-c~(1/2),CD=a-c~(1/2) ,BE=EC=c~(2/1)则AE=,b~(1/2),DE=a~(1/2),连AD,则 S梯形ABCD=S△ABB+S△CDE+S△AED。
According to certain conditions if you can construct a right-angled triangle, use the nature of the right-angled triangle to make some unreasonable inequalities get intuitive and clever proofs. Here are some examples to illustrate: Example 1 Let a≥c,b≥c,c ≥ 0, verify and confirm the conditions for the establishment of the equal sign. Proof: As shown in Fig. 1, Rt △ ABE and Rt △ ECD are made on the segment BC of length c ~ (1/2) so that AB = bc ~ (1/2), CD = ac ~ (1/2), BE=EC=c~(2/1) then AE=, b~(1/2), DE=a~(1/2), even AD, then S trapezoid ABCD=S△ABB+S△CDE+S △AED.