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正如《中学数学》86年第10期上《圆外切四边形判定定理的证明》一文中所说:这定理的证明常采用反证法。笔者在教学中发现还可以有如下直接证明的方法,过程并不繁复,现介绍如下。已知四边形ABCD中,AD+BC=AB+CD,求证四边形ABCD存在内切圆。证明如四边形ABCD是菱形,则结论显然成立,若ABCD不是菱形,不失一般性,设AB>AD,则必有BC>CD(图下)。
As stated in the article “The proof of the circle outer tangent decision theorem” in the “Middle School Mathematics” issue No. 10, 86: the proof of this theorem often uses counter-evidence. The author found in the teaching can also have the following direct proof method, the process is not complicated, is introduced as follows. It is known that in the quadrilateral ABCD, AD+BC=AB+CD confirms that there is an inscribed circle in the quadrilateral ABCD. Prove that if the quadrangle ABCD is diamond, the conclusion is obviously established. If ABCD is not rhombic, without loss of generality, suppose AB>AD, there must be BC>CD (lower figure).