Try this beautiful problem from Geometry: Circle
Circles with centers $A$ and $B$ have radii 3 and 8 , respectively. A common internal tangent intersects the circles at $C$ and $D$, respectively. Lines $A B$ and $C D$ intersect at $E,$ and $A E=5 .$ What is $C D ?$
,
i
Geometry
Circle
Tangents
Answer: $ \frac{44}{3}$
AMC-10 (2006) Problem 23
Pre College Mathematics
Given that Circles with centers $A$ and $B$ have radii 3 and 8 and $A E=5 .$.we have to find out \(CD\).So join \(BC\) and \(AD\).then clearly \(\triangle BCE\) and \(\triangle ADE\) are Right-Triangle(as \(CD\) is the common tangent ).Now \(\triangle BCE\) and \(\triangle ADE\) are similar.Can you proof \(\triangle BCE\) and \(\triangle ADE\)?
Can you now finish the problem ..........
$\angle A E D$ and $\angle B E C$ are vertical angles so they are congruent, as are angles $\angle A D E$ and $\angle B C E$ (both are right angles because the radius and tangent line at a point on a circle are always perpendicular). Thus, $\triangle A C E \sim \triangle B D E$.
By the Pythagorean Theorem, line segment \(DE=4\)
Therefore from the similarity we can say that \(\frac{D E}{A D}=\frac{C E}{B C} \Rightarrow \frac{4}{3}=\frac{C E}{8}\) .
Therefore \(C E=\frac{32}{3}\)
can you finish the problem........
Therefore \(CD=CE+DE=4+\frac{32}{3}=\frac{44}{3}\)
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