Competition problem — Geometry — Maths Olympiad Prep

Library / /209 of 377

Geometry Difficulty 5.1 AIME, harder Prove it United States

Problem:

Circle $B$ has radius $6 \sqrt{7}$ . Circle $A$ , centered at point $C$ , has radius $\sqrt{7}$ 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 $X Y$ is also tangent to $A$ . Find the area contained by the boundary of $L$ .

Solution

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 $\sqrt{(R-2 r) R}$ , where $R, r$ are the circumradius and inradius, respectively. Thus the locus of $C$ is a circle concentric to $B$ with radius $2 \sqrt{42}$ . The conclusion follows.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.