HMMT 二月 2023 · 几何 · 第 7 题

HMMT February 2023 — Geometry — Problem 7

Quadrilateral ABCD is inscribed in circle Γ. Segments AC and BD intersect at E . Circle γ passes through E and is tangent to Γ at A . Suppose that the circumcircle of triangle BCE is tangent to γ at E and is tangent to line CD at C . Suppose that Γ has radius 3 and γ has radius 2. Compute BD . ◦

解析

Quadrilateral ABCD is inscribed in circle Γ. Segments AC and BD intersect at E . Circle γ passes through E and is tangent to Γ at A . Suppose that the circumcircle of triangle BCE is tangent to γ at E and is tangent to line CD at C . Suppose that Γ has radius 3 and γ has radius 2. Compute BD . Proposed by: Eric Shen, Luke Robitaille √ 9 21 Answer: 7 Solution: The key observation is that △ ACD is equilateral. This is proven in two steps. • From tangency at C , we have ∠ DCA = ∠ DCE = ∠ EBC = ∠ DBC = ∠ DAC, implying that CA = CD . • Consider the common tangent of γ and Γ at A . By homothety at E , this line is parallel to the tangent of ⊙ ( EBC ) at C , which is line CD . This implies that AC = AD . Once we have this, compute √ ◦ AC = 2 R · sin 60 = 3 3 Γ √ ◦ AE = 2 R · sin 60 = 2 3 γ √ There are now many ways to finish. One way is to use Stewart’s theorem on △ ADC to get ED = 21, √ √ AE · EC 2 21 9 21 then use Power of Point to get EB = = . The final answer is BD = BE + ED = . ED 7 7 ◦