The radius ОМ of the circle with the center О divides the chord AB in half.

The radius ОМ of the circle with the center О divides the chord AB in half. Prove that the tangent drawn through the point M is parallel to the chord AB.

Let a circle be given with the center located at point O. A chord AB is drawn in the circle. By condition, the radius OM divides this chord in half at point P, that is, AR = PB. Through point M, a tangent to the circle is drawn – the straight line CК. By the property of tangents, the segment OM is perpendicular to the CК. In addition, OM is perpendicular to AB, since in an isosceles triangle AOB (AO = OB = R), the median, by the property of the median (AR = PB by condition), is the bisector and height. Two straight lines AB and SC, perpendicular to the third straight line OM, are parallel to each other AB || SK. Q.E.D.