凹四边形一个结论的多种证法及灵活运用

• 作 者: 华兴恒
• 机 构: 安徽省灵璧县黄湾中学,
• 出 处: 数学大世界(初中版)
• 发表日期: 2013年05期

结论如图1所示,在凹四边形ABCD中,∠BDC=∠A+∠B+∠C.分析对于上述结论,利用三角形外角的性质和平行线的性质可以探索出多种添加辅助线的方法予以证明.证法1连接AD并延长到E,如图2所示,由三角形的外角性质很容易证明.∵∠CDE=∠C+∠CAD,∠EDB=∠DAB+∠B.而∠CDE+∠EDB=∠BDC,∠CAD+∠DBAD=∠A∴∠BDC=∠A+∠B+∠C.证法2连结BC,如图3所示,由三角形的内角和

Conclusion As shown in Fig. 1, ∠BDC = ∠A + ∠B + ∠C in the concave quadrilateral ABCD. For the above conclusion, we can find many ways to add auxiliary lines by using the properties of triangular outer corners and parallel lines .Acquisition 1 connected AD and extended to E, as shown in Figure 2, by the nature of the triangle outside the triangle is easy to prove.CDE = ∠C + ∠CAD, ∠EDB = ∠DAB + ∠B. And ∠CDE + ∠EDB = ∠ BDC, ∠ CAD + ∠DBAD = ∠A ∴∠BDC = ∠A + ∠B + ∠C. Syndrome 2 links BC, as shown in Figure 3, by the internal angle of the triangle and