In the figure,$DE \parallel AC$ and $DF \parallel AE$. Prove that $\frac{BF}{FE} = \frac{BE}{EC}$.

In the figure,$D$ is a point on side $BC$ of $\Delta ABC$ such that $\frac{BD}{CD} = \frac{AB}{AC}$. Prove that $AD$ is the bisector of $\angle BAC$.

In the figure,$OA \cdot OB = OC \cdot OD$. Show that $\angle A = \angle C$ and $\angle B = \angle D$.

Nazima is fly fishing in a stream. The tip of her fishing rod is $1.8 \, m$ above the surface of the water and the fly at the end of the string rests on the water $2.4 \, m$ away from a point directly under the tip of the rod. Assuming that her string (from the tip of her rod to the fly) is taut,how much string does she have out? If she pulls in the string at the rate of $5 \, cm$ per second,what will be the horizontal distance of the fly from her after $12 \, seconds$?

In the figure,$ABD$ is a triangle right-angled at $A$ and $AC \perp BD$. Show that $AC^{2} = BC \cdot DC$.

Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.