Proof of Secant-Tangent
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 D is a tangent point and also 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