Triangle Drawn to a Circle

Geometry Level 2

Given circle $\Gamma$ , point $A$ is chosen outside of $\Gamma$ . Tangents $AB$ and $AC$ to $\Gamma$ are drawn. $K$ is a point on the circumference of $\Gamma$ contained within $ABC$ . $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $DKE$ is a straight line and tangent to $\Gamma$ . If $AB = 19$ , what is the perimeter of triangle $ADE$ ?

The answer is 38.

1 solution

Arron Kau Staff
May 13, 2014

Recall that the lengths of tangents from a point to a circle are equal. Since $AB$ and $AC$ are tangent to $\Gamma$ , we have $AB = AC$ . Similarly, $DK = DB$ and $KE = EC$ . Thus the perimeter of triangle $ADE$ is equal to $\begin{aligned} AD + AE + DE &= AD + AE + DK + KE \\ &= AD + AE + DB + EC \\ &= AB + AC \\ &= 2AB \\ &= 2\times 19 \\ &= 38 \\ \end{aligned}$

Triangle Drawn to a Circle

The answer is 38.

1 solution

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