HMMT November

Geometry Difficulty 4.7 AIME Find the answer

Circle $O$ has chord $A B$ . A circle is tangent to $O$ at $T$ and tangent to $A B$ at $X$ such that $A X=2 X B$ . What is $\text{A}$ T}{B T}?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $T X$ meet circle $O$ again at $Y$ . Since the homothety centered at $T$ takes $X$ to $Y$ also takes $A B$ to the tangent line of circle $O$ passing through $Y$ , we have $Y$ is the midpoint of arc $A B$ . This means that $\angle A T Y=\angle Y T B$ . By the Angle Bisector Theorem, $\frac{A T}{B T}=\frac{A X}{B X}=2$ .

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