Vertex B of rhombus ABCD is the center of a circle whose radius is half of the diagonal BD.

Vertex B of rhombus ABCD is the center of a circle whose radius is half of the diagonal BD. Prove that line AC is tangent to the circle.

By the property of the diagonals of a rhombus, the point of their intersection divides the diagonals in half and forms a right angle at the point of intersection. Then the angle BOC = 90, and BO = DO.

Since the radius of the circle is R = BO, and point O is the point of intersection of the diagonals, then point O is the common point of the circle and the diagonal AC, and since the angle BOC = 90, then AC is tangent to the circle, which was required to prove.