Why Does Ptolemy Start With :-P

Geometry Level 2

\(ABCD\) is a cyclic quadrilateral with \(\displaystyle \overline{AB}=11\) and \(\displaystyle \overline{CD}=19\). \(P\) and \(Q\) are points on \(\overline{AB}\) and \( \overline{CD}\), respectively, such that \(\displaystyle \overline{AP}=6\), \(\displaystyle \overline{DQ}=7\), and \(\displaystyle \overline{PQ}=27.\) Determine the length of the line segment formed when \(\displaystyle \overline{PQ}\) is extended from both sides until it reaches the circle.

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