Geometry
# Tangent and Secant Lines

\(ABCD\) 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.

**Note**: The image is not drawn to scale.

You are currently located on point \((0,0)\) and you want to get on point \((54,18)\). However, there are some annoying circular objects in the way! They are defined by \[x^2+y^2-18x-18y+81=0\] \[x^2+y^2-90x-18y+2025=0\]

If you cannot walk through these annoying circular objects, then the shortest possible path possible to point \((54,18)\) can be expressed as \[a+b\sqrt{c}+d\pi\] for positive integers \(a,b,c,d\) with \(c\) square-free. What is \(a+b+c+d\)?

In \(\triangle ABC,\) \(AB=6 \) , \(BC = 4\) and \( AC=8.\) A segment parallel to \( \overline{BC}\) and tangent to the incircle of \(\triangle ABC\) intersects \(\overline{AB}\) at \(M\) and \(\overline{AC}\) at \(N\).

If \(MN=\dfrac{a}{b}\), where \(a\) and \(b\) are co-prime positive integers, what is the value of \(a+b\) ?

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