Radius of a Circle, Given Tangent

Geometry Level 5

$P$ is a point outside of circle $\Gamma$ . The tangent from $P$ to $\Gamma$ touches at $A$ . A line from $P$ intersects $\Gamma$ at $B$ and $C$ such that $\angle ACP = 120^\circ$ . If $AC = 16$ and $AP = 19$ , then the radius of $\Gamma$ can be expressed as $\frac {a \sqrt{b} }{c}$ , where $b$ is an integer not divisible by the square of any prime and $a, c$ are coprime positive integers. What is $a + b + c$ ?

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Radius of a Circle, Given Tangent

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, interactive explorations.

Used and loved by over 7 million people

Learn from a vibrant community of students and enthusiasts, including olympiad champions, researchers, and professionals.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.