In the quadrilateral ABCD, the diagonal AC is perpendicular to the diagonal BD and divides it in half. Prove that triangles ABC

In the quadrilateral ABCD, the diagonal AC is perpendicular to the diagonal BD and divides it in half. Prove that triangles ABC and ADC are equal and AC is the bisector of the angle BAD.

Let’s deal with this problem.
Given:
ABCD – quadrilateral
AC⊥BD.
Prove:
ΔABC = ADC
AC is the bisector of ∠BAD.

Decision:
We denote the point of intersection of the diagonals by O, then BO = AO (by condition)
AO- common leg for right-angled triangles ABO and AOD. Therefore, these triangles are equal, which means AB = AD. Similarly, the equality of triangles BOC = DOC and output BC = CD. Consider triangles ABC and ADC:
AB = AD, BC = CD
AC is a common side, so triangles ABC and ADC are equal (on three sides).
Q.E.D.