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PUMaC 2023 · 几何（B 组） · 第 1 题

PUMaC 2023 — Geometry (Division B) — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. Rectangle ABCD has AB = 24 and BC = 7. Let d be the distance between the centers of the 2 incircles of △ ABC and △ CDA . Find d .
解析
  1. Rectangle ABCD has AB = 24 and BC = 7. Let d be the distance between the centers of the 2 incircles of △ ABC and △ CDA . Find d . Proposed by Atharva Pathak Answer: 325 1 · 7 · 24 2 Since △ ABC has sidelengths 7 , 24 , 25, the identity K = rs implies r = = 3. Since 7+24+25 2 △ ABC and △ CDA are congruent, the inradius of △ CDA is also 3. Thus the horizontal distance between the incenters is 24 − 3 − 3 = 18, and the vertical distance between the 2 2 2 incenters is 7 − 3 − 3 = 1. By Pythagoras, our answer is d = 18 + 1 = 325.