Tangent Circles Make Great Triangles!

Geometry Level 3

Let $\omega_1$ and $\omega_2$ be two externally tangent circles at $T$ with $\omega_1$ smaller than $\omega_2$ . Let their common external tangents intersect at $S$ . Let a line through $S$ intersect circles $\omega_1$ and $\omega_2$ at $A,B,C,D$ in that order. Given that $TC = 3$ and $TA = 4$ , find $AC$ .

The answer is 5.

1 solution

Alan Yan
Oct 17, 2015

Connect $S$ with $T$ and call the intersection with $\omega_1$ , $T'$ . Positive homothety centered at $S$ maps $A \rightarrow C$ and $T' \rightarrow T$ . This implies that $AT' || TC \implies \angle CTA = 90^{\circ}.$

Thus, $\triangle ATC$ is a right triangle and Pythagorean theorem, you get that $AC = \boxed{5}$ .

will you pls put a diagram

Riddhesh Deshmukh - 5 years, 7 months ago

I will! \8char

Alan Yan - 5 years, 7 months ago

Tangent Circles Make Great Triangles!

The answer is 5.

1 solution

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