Line segments $AB$ and $CD$ bisect each other at $O$. Let's prove that $AC$ and $BD$ are parallel. Let's write what kind of quadrilateral is $ABCD$.

E and F are two points of two parallel straight lines AB and CD respectively. O is the midpoint of line segment EF. We draw a straight line passing through O which intersect AB and CD at P and Q respectively. Let's prove that O bisects the line segment PQ.

If we produce the base of an isosceles triangle on both sides then two exterior angles are formed. Let's prove that they are equal in measurement.

Let's prove that the medians of the equilateral triangle are equal.

In a trapezium ABCD, AD||BC and angle ABC= angle BCD. Let's prove that ABCD is an isosceles trapezium.

$AB$ is the hypotenuse of right isosceles triangle $ABC$. $AD$ is the bisector of angle $\angle BAC$ and $AD$ intersects $BC$ at $D$. Let's prove that $AC+CD=AB$

Two isosceles triangles $ABC$ and $DBC$ whose $AB=AC, DB=DC$ and they are situated on the opposite side of $BC$. Let's prove that $AD$ bisects $BC$ perpendicularly.

Two line segments $PQ$ and $RS$ intersect each other at $X$ in such a way that $XP=XR$ and $\angle PSX=\angle RQX$. Let's prove that $△PXS\cong △RQX$.