In a circle centered at a point o, the radius OM is perpendicular to the chord AB. Prove that the resulting chord segments are equal.

Point H is the intersection point of the chord AB and the radius OM. Since, by condition, OM is perpendicular to AB, the triangles AOН and BOН are rectangular. In right-angled triangles AOН and ВOН, the leg OH is common, and the hypotenuse OA = OB as the radii of the circle, then the triangles AON and ВOН are equal in the third sign, in the leg and hypotenuse.
Then AH = BH as similar sides of equal triangles, as required to prove.