Geometry - Using Formulas #1

The formula used for solving this problem is Ptolemy's Theorem .

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Since $\angle \text{ABC}=180^{\circ}-\angle \text{CDA}$ , we know that quadrilateral $\text{ABCD}$ is inscribed in a circle.

Therefore, according to Ptolemy's Theorem , $\overline{\text{AB}}\cdot\overline{\text{CD}}+\overline{\text{BC}}\cdot\overline{\text{DA}}=\overline{\text{AC}}\cdot\overline{\text{BD}}$ .

Let $\overline{\text{AB}}\cdot\overline{\text{CD}}=a$ and $\overline{\text{BC}}\cdot\overline{\text{DA}}=b$ .

From the given conditions we can see that $ab=14$ and $a+b=8$ .

If we let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2-8x+14=0$ , possible values for $\dfrac{\overline{\text{AB}}}{\overline{\text{BC}}}\cdot\dfrac{\overline{\text{CD}}}{\overline{\text{DA}}}=\dfrac{a}{b}$ are $\dfrac{\beta}{\alpha}$ and $\dfrac{\alpha}{\beta}$ .

Also, $\alpha+\beta=8$ and $\alpha\beta=14$ .

Therefore, the sum we're trying to find is $\dfrac{\beta}{\alpha}+\dfrac{\alpha}{\beta}=\dfrac{(\alpha+\beta)^2-2\alpha\beta}{\alpha\beta}=\boxed{\dfrac{18}{7}=2.57142857...}.$