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初中《几何》第二册第29页20题:求证在圆内接四边形ABCD中,AB·CD+BC·AD=AC·BD。这是一道有用的习题.利用它的结论处理有些问题较为方便。因此,我建议教师们在教学中不可忽视它,可让同学们记住它的结论,证某些题可直接运用,现举两例说明它的作用。例1 已知P是正方形ABCD外接圆AD劣弧上一点,求证:(1)(PB+PD)/PC=2~(1/2):(2)(PB-PD)/PA=2~(1/2);(3)PB~2-PD~2=2 PA·PC。证明:(1)在圆内接四边形PBCD中,有PB·CD+PD·BC=BD·PC。
Junior high school “Geometry” Volume II, page 29, 20 questions: verification in the circle inscribed in quadrilateral ABCD, AB CD +BC AD = AC BD. This is a useful exercise. It is more convenient to use its conclusions to deal with some issues. Therefore, I suggest that teachers should not ignore it in teaching. They can let students remember its conclusions and prove that certain questions can be used directly. Now two examples are given to illustrate its role. Example 1 It is known that P is a point on the inferior arc of a square ABCD circumcircle AD. Proof: (1) (PB+PD)/PC=2~(1/2):(2)(PB-PD)/PA=2~ (1/2); (3) PB~2-PD~2=2 PA·PC. Prove that: (1) Among the PBCDs that are in quadratic circles, there are PB·CD+PD·BC=BD·PC.