Prove that the segments of the tangents to the circle drawn from one point are equal and make equal angles

Prove that the segments of the tangents to the circle drawn from one point are equal and make equal angles with the straight line passing through this point and the center of the circle.

Let’s build the radii of the OB and OС.

The radii OB and OС are drawn to the points of tangency B and C of tangents AB and AC from the center of the circle, then the radii OB and OС are perpendicular to tangent AB and AC, and then the triangles AOC and AOB are rectangular.

In right-angled triangles AOB and AOС, the hypotenuse of AO is common, leg OB = O = R, then triangles AOB and AOC are equal in leg and hypotenuse, the first sign of equality of right-angled triangles. Then the angle ОАВ = ОАС, and the leg AB = АС, which was required to be proved.