Show that the diagonals of a rhombus divide it into four congruent triangles.

Extend $AB$ to $E$. find $\angle CBE$. What do you notice? What kind of angles are $ABC$ and $CBE$?

Show that the diagonals of a square are equal and right bisectors of each other.

$\mathrm{ABC}$ is an isosceles triangle in which $\text{AB=AC}$. $\mathrm{AD}$ bisects exterior angle $\mathrm{QAC}$ and $\mathrm{CD}\left|\right|\mathrm{BA}$ as shown in the figure. Show that $\angle \mathrm{DAC}=\angle \mathrm{BCA}$

$ABC$ is a triangle right-angled at $C$. A line through the midpoint $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that

$D$ is the midpoint of $AC$.

$ABC$ is a triangle right-angled at $C$. A line through the midpoint $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that

$MD\perp AC$

$ABC$ is a triangle right angled at $C$. A line through the midpoint $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that

$CM=MA=\frac{1}{2}AB$