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A tangent to a circle is a line intersecting the circle at exactly one point. Can you prove that the line from the center of the circle to the point of tangency is perpendicular to the tangent line?

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\(ABCD\) is a Cyclic Quadrilateral, with \(\displaystyle \overline{AB}=11\), \(\displaystyle \overline{CD}=19\). \(P\) and \(Q\) are points on \(\overline{AB}\) and \( \overline{CD}\) such that \(\displaystyle \overline{AP}=6\) and \(\displaystyle \overline{DQ}=7\). We have \(\displaystyle \overline{PQ}=27.\) 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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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 \(a+b+c+d\)?

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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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