Geometry
# Tangent and Secant Lines

As shown in the above diagram, triangle \(ABC\) inscribed in a circle is an isosceles triangle with \(\lvert{\overline{AB}}\rvert=\lvert{\overline{AC}}\rvert,\) where \(\lvert{\overline{AB}}\rvert\) denotes the length of \({\overline{AB}}.\) Given the following two lengths: \[\lvert{\overline{AP}}\rvert=11 \text{ and } \lvert{\overline{PQ}}\rvert=5,\] what is \({\lvert\overline{AB}\rvert}^2?\)

**Note:** The above diagram is not drawn to scale.

The above diagram illustrates a \(17\) meters by \(10\) meters rectangular farmland \(ABCD.\) The farmland contains a circular well \(O\) with radius \(4\) meters at its upper left corner, which is tangent to \(\overline{AB}\) and \(\overline{AD}.\) If the farmer wants to build a straight fence dividing the land into two areas in the following way, what should be the length of \(\overline{FD}\) (in meters):

- \(\angle FEC = 45 ^\circ\)
- the well goes to the left of the fence
- make the area of \(FECD\) as large as possible?