HMMT 二月 2013 · 几何 · 第 6 题

HMMT February 2013 — Geometry — Problem 6

Let ABCD be a quadrilateral such that ∠ ABC = ∠ CDA = 90 , and BC = 7. Let E and F be on BD such that AE and CF are perpendicular to BD . Suppose that BE = 3. Determine the product of the smallest and largest possible lengths of DF . ◦ ◦

解析

Let ABCD be a quadrilateral such that ∠ ABC = ∠ CDA = 90 , and BC = 7. Let E and F be on BD such that AE and CF are perpendicular to BD . Suppose that BE = 3. Determine the product of the smallest and largest possible lengths of DF . Answer: 9 By inscribed angles, ∠ CDB = ∠ CAB , and ∠ ABD = ∠ ACD . By definition, ∠ AEB = ∠ CDA = ∠ ABC = ∠ CF A . Thus, △ ABE ∼ △ ADC and △ CDF ∼ △ CAB . This shows that BE CD DF AB = and = AB CA CD BD Based on the previous two equations, it is sufficient to conclude that 3 = EB = F D . Thus, F D must equal to 3, and the product of its largest and smallest length is 9. ◦ ◦