In the figure,altitudes $AD$ and $CE$ of $\Delta ABC$ intersect each other at the point $P$. Show that $\Delta AEP \sim \Delta CDP$.

$ABC$ is an isosceles triangle with $AC = BC$. If $AB^2 = 2 AC^2$,prove that $ABC$ is a right triangle.

In the figure,the line segment $XY$ is parallel to side $AC$ of $\Delta ABC$ and it divides the triangle into two parts of equal areas. Find the ratio $\frac{AX}{AB}$.

An aeroplane leaves an airport and flies due north at a speed of $1000\, km/h$. At the same time,another aeroplane leaves the same airport and flies due west at a speed of $1200\, km/h$. How far apart will be the two planes after $1 \frac{1}{2}$ hours?

In the figure,two chords $AB$ and $CD$ of a circle intersect each other at point $P$ (when produced) outside the circle. Prove that $\Delta PAC \sim \Delta PDB$.

$D$ is a point on the side $BC$ of a triangle $ABC$ such that $\angle ADC = \angle BAC$. Show that $CA^2 = CB \cdot CD$.