Prove that the diameter drawn through the midpoint of a chord of the same circle other than the diameter

Prove that the diameter drawn through the midpoint of a chord of the same circle other than the diameter is perpendicular to this chord.

Let’s construct the radii OC and OD to the edges of the chord CD.

In the OCD triangle, OС = OD = R, therefore, the OCD triangle is isosceles.

By condition, CH = DH, then the OH segment is the median of the OCD triangle.

The median of an isosceles triangle, drawn to its base, is also its height and bisector, then OH is perpendicular to CD, and then the diameter AB is perpendicular to the chord CD, which was required to be proved.