HMMT 十一月 2011 · GEN 赛 · 第 9 题
HMMT November 2011 — GEN Round — Problem 9
题目详情
英文原题
- [ 7 ] Let P and Q be points on line l with P Q = 12. Two circles, ω and Ω, are both tangent to l at Pand are externally tangent to each other. A line through Q intersects ω at A and B , with A closer to
Q than B , such that AB = 10. Similarly, another line through Q intersects Ω at C and D , with Ccloser to Q than D , such that CD = 7. Find the ratio AD/BC . 7
解析
英文解析
- [ 7 ] Let P and Q be points on line l with P Q = 12. Two circles, ω and Ω, are both tangent to l at Pand are externally tangent to each other. A line through Q intersects ω at A and B , with A closer to
Q than B , such that AB = 10. Similarly, another line through Q intersects Ω at C and D , with Ccloser to Q than D , such that CD = 7. Find the ratio AD/BC .
8 2
Answer: We first apply the Power of a Point theorem repeatedly. Note that QA · QB = QP =
QC · QD . Substituting in our known values, we obtain QA ( QA + 10) = 12 = QC ( QC + 7). Solving 29
these quadratics, we get that QA = 8 and QC = 9.
AQ CQ
We can see that = and that ∠ AQD = ∠ CQB , so QAD ∼ QCB . (Alternatively, going
DQ BQback to the equality QA · QB = QC · QD , we realize that this is just a Power of a Point theorem onthe quadrilateral ABDC , and so this quadrilateral is cyclic. This implies that ∠ ADQ = ∠ ADC =
AD 8AQ
∠ ABC = ∠ QBC .) Thus, = = .
BC QC 9
7D
9CΩ
P 12Qω
A8
B10 7