Geometry
# Tangent and Secant Lines

$ABC$ 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 ${\lvert\overline{AB}\rvert}^2?$

As shown in the above diagram, triangle**Note:** The above diagram is not drawn to scale.

$17$ 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):

The above diagram illustrates a- $\angle FEC = 45 ^\circ$
- the well goes to the left of the fence
- make the area of $FECD$ as large as possible?

$ABCD$ that measures $30$ meters by $25$ meters. There are two wells, one at the upper left corner and another at the lower right corner, with the same radius. Both wells are tangent to the borders of the land. When you put a fence $EF$ that is tangent to both wells, as shown in the above diagram, you find $\angle EFB=60^\circ.$ What is the radius of the two wells?

You have a piece of rectangular land