# quadrilateral and circle - geometry

Can anyone help me on this problem? I wonder what is an easy way to solve the problem. Thank you very much!

Quadrilateral ABCD is circumscribed about a circle. Three sides of the quadrilateral are 9, 17, and 12. Find the length of side AD. • Hint: The two lines segments from say $A$ to the points of tangency of the tangents through $A$ are of equal length. Call their length $x$. Continue. – André Nicolas Mar 21 '16 at 1:49
• @Andre Nicolas. Could you please give one more hint? I don't know how to continue. – N.S.JOHN Mar 21 '16 at 2:23

## 1 Answer

Temporarily assume that $AB=9$, $BC=17$, and $CD=12$. We will revisit that later.

Let the lengths of the two line segments through $A$ that are tangent to the circle and end at the points of tangency be $w$. Let the corresponding lengths through $B$ be $x$, the ones through $C$ be $y$, and through $D$ be $z$.

Draw a picture, and label the line segments as mentioned above. So you will write $8$ labels.

We have the following equations:

$w+x=9$

$x+y=17$

$y+z=12$.

Note that we want to find $w+z$.

We have $$w+z=(w+x)+(y+z)-(x+y).$$

For our choice of lengths for the various sides, this gives $w+z=21-17=4$.

But there are other choices for which sides have lengths $9$, $17$, and $12$. These give other answers.

The question as written in the OP does not specify which sides have which lengths. Perhaps the original question does. If not, there are $3$ possible answers for the length of $AD$.