PUMaC 2020 · 几何（B 组） · 第 8 题

PUMaC 2020 — Geometry (Division B) — Problem 8

Triangle ABC has side lengths 13, 14, and 15. Let E be the ellipse that encloses the smallest √ a bπ area which passes through A , B , and C . The area of E is of the form , where a and c are c coprime and b has no square factors. Find a + b + c . 1

解析

Triangle ABC has side lengths 13, 14, and 15. Let E be the ellipse that encloses the smallest √ a bπ area which passes through A , B , and C . The area of E is of the form , where a and c are c coprime and b has no square factors. Find a + b + c . Proposed by: Daniel Carter Answer: 118 Let T be an affine transformation that sends an equilateral triangle with side length 1 to triangle ABC . Affine transformations preserve the ratios of areas, so the smallest such ellipse for the equilateral triangle will be sent to E by T . It is clear by inspection that the smallest area ellipse for the equilateral triangle is its circumcircle. The circumcircle of an equilateral triangle √ 4 3 π has area times the area of the triangle, and the area of ABC is 84 (found via Heron’s 9 √ √ 4 3 π 112 3 π formula), so the area of the E is · 84 = . Thus the answer is 112 + 3 + 3 = 118. 9 3 3