# Proof of Secant-Tangent

- Author:
- Adam

In the diagram below we'll prove that

Draw the triangle inscribed in the circle. Call the angle measure and therefore because this is an inscribed angle, we get that the arc measure shown in red, .

If we now use the fact that the arc

If we next use the fact that is an inscribed angle at a point of tangency, then we get to infer that or after simplifying .
But now we can notice that we have two triangles with two matching angles. The two triangles are and . The two matching angles are and . This means that we get to infer by Angle-Angle Theorem that the triangles are similar, but we still need to state the similarity relationship in the right order. It is

*D*is a tangent point and also