What theorem relates the length of tangents drawn from the same external point to a circle?

The theorem that relates to the length of tangents drawn from the same external point to a circle states that the lengths of the two tangents from that point to the points of tangency on the circle are equal. This stems from the properties of a circle and the geometry of tangents.

When a tangent is drawn from an external point to a circle, it forms a right angle with the radius at the point of tangency. By drawing a line from the external point to the center of the circle and then connecting the two points where the tangents touch the circle, you create two right triangles. These triangles share a common side (the line connecting the external point to the center), and since they both have a tangent line that meets the circle perpendicularly, they are congruent by the Hypotenuse-Leg theorem.

Thus, because these triangles are congruent, the lengths of the two tangents from the external point to the points of contact with the circle must be equal. This theorem is useful in various geometric proofs and problems involving circles.