连接AC，如图：

∵ $∠ D = 90^{\circ}$ ， $A D = C D$

∴ $∠ A C D = ∠ D A C = 45^{\circ}$

∵ $A D / / B C$

∴ $∠ D = ∠ D C E = 90^{\circ}$

∴ $∠ A C E = 45^{\circ} + 90^{\circ} = 135^{\circ}$

又∵ $∠ A C D + ∠ A C F = 180^{\circ}$

∴ $∠ A C F = 135^{\circ}$

∴ $∠ A C E = ∠ A C F$

∵ $∠ A C F + ∠ C A F + ∠ C F A = 180^{\circ}$

∴ $∠ C A F + ∠ C F A = 180^{\circ} - 135^{\circ} = 45^{\circ}$

∵ $∠ F A E = 45^{\circ}$

∴ $∠ C A F + ∠ C A E = 45^{\circ}$

∴ $∠ C A E = ∠ C F A$

∴ $△ C A F ∽ △ C E A$

∴ $\frac{A C}{C E} = \frac{C F}{A C}$

∴ $A C^{2} = C E \cdot C F$

上海市浦东新区2020-2021学年九年级上学期数学10月月考试卷