HMMT 二月 2009 · 几何 · 第 5 题

HMMT February 2009 — Geometry — Problem 5

[ 4 ] Circle B has radius 6 7. Circle A , centered at point C , has radius 7 and is contained in B . Let L be the locus of centers C such that there exists a point D on the boundary of B with the following property: if the tangents from D to circle A intersect circle B again at X and Y , then XY is also tangent to A . Find the area contained by the boundary of L .

解析

[ 4 ] Circle B has radius 6 7. Circle A , centered at point C , has radius 7 and is contained in B . Let L be the locus of centers C such that there exists a point D on the boundary of B with the following property: if the tangents from D to circle A intersect circle B again at X and Y , then XY is also tangent to A . Find the area contained by the boundary of L . Answer: 168 π Solution: The conditions imply that there exists a triangle such that B is the circumcircle and A is the incircle for the position of A . The distance between the circumcenter and incenter is given by √ ( R − 2 r ) R , where R , r are the circumradius and inradius, respectively. Thus the locus of C is a √ circle concentric to B with radius 2 42. The conclusion follows.