PUMaC 2008 · 几何（B 组） · 第 10 题

PUMaC 2008 — Geometry (Division B) — Problem 10

(7 points) A cuboctahedron is the convex hull of (smallest convex set containing) the 12 points ( ± 1 , ± 1 , 0), ( ± 1 , 0 , ± 1), (0 , ± 1 , ± 1). Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the figure. The area of that set is the solid angle of the figure as viewed from the origin.) 2

解析

Find the coordinates of the point in the plane at which the sum of the distances from it to the √ three points (0 , 0), (2 , 0), (0 , 3) is minimal. √ 5 3 3 ( ANS: ( , ). It is well known that the point we are looking for is the Fermat Point, the point 13 13 ◦ F with the property that when we draw the segments F A , F B , and F C , they make three 120 angles at F . We can find this point by intersecting the circumcircles of each of three equilateral triangles mounted on each edge of the original triangle. Finding the equations of two of these circles is easy given the coordinates of the triangle. We can also easily check that the given √ ◦ answer (5 / 13 , 3 3 / 13) works by making sure the specified angles are indeed 120 . We know ◦ 1 cos 120 = − . Thus we can check angles by checking dot products. This verification is left to the 2 reader. CB: JP)