HMMT 二月 2010 · 几何 · 第 9 题

英文解析

9. [ 7 ] Let ABCD be a quadrilateral with an inscribed circle centered at I . Let CI intersect AB at E . If
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   ∠ IDE = 35 , ∠ ABC = 70 , and ∠ BCD = 60 , then what are all possible measures of ∠ CDA ?
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   Answer: 70 and 160
   Arbitrarily defining B and C determines I and E up to reflections across BC . D lies on both the circle
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   determined by ∠ EDI = 35 and the line through C tangent to the circle (and on the opposite side of
   B ); since the intersection of a line and a circle has at most two points, there are only two cases for
   ABCD . The diagram below on the left shows the construction made in this solution, containing bothcases. The diagram below on the right shows only the degenerate case.
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   B D
   E ADA
   I I
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   70 70
   60 60
   B C B C
   Geometry Subject Test
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   Reflect B across EC to B ; then CB = CB . Since BA and BC are tangent to the circle centered
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   at I , IB is the angle bisector of ∠ ABC . Then ∠ IBE = ∠ IB E = 35 . If B = D , then ∠ ADC =
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   ∠ EB C = 70 . Otherwise, since ∠ IB E = 35 = ∠ IDE (given), EB DI is a cyclic quadrilateral.
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   Then ∠ IED = ∠ IB D = 35 and ∠ BCI = ∠ ECD = 30 , so 4 CED ∼ 4 CBI .
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   Since ∠ CID is exterior to 4 DIE , ∠ CID = ∠ IDE + ∠ DEI = 70 . Then 4 CDI ∼ 4 CEB . Because
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   EB DI is cyclic, ∠ IDC = ∠ IEB = ∠ IEB = 180 − 70 − 30 = 80 . Then ∠ ADC = 2 ∠ IDC = 160 .
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   Thus, the two possible measures are 70 and 160 .