What is true about tangents sharing a common vertex?

Tangents that share a common vertex are indeed equal in length when they are drawn from that vertex to points on the circle. This property is a result of the fact that two tangents drawn from an external point to a circle will always form line segments that are equal to each other. This stems from the congruent triangles formed by the radius at the point of tangency and the segments connecting the external point to those points of tangency.

Additionally, these tangents create an angle of 90 degrees with the radius at the point of tangency, establishing the right-angle relationship, but that is not the reason for the equality of lengths. Each tangent line is perpendicular to the radius of the circle at the point where it touches the circle, reinforcing the concept of tangents in geometry. This characteristic leads to the reliable conclusion that both tangents are congruent.

The other statements regarding tangents sharing a common vertex either do not necessarily hold true or only contribute to the understanding of their geometric properties but do not confirm their lengths as being equal. The equality of the lengths is the most critical aspect in relation to the tangents in this context.