Question

Condition for tangent to be perpendicular to each other in a circle

Answer 1

Proof

Let  be the point of tangency,  be the center of the circle, and  be the foot of the altitude from  to the tangency line. Suppose that  and  are different points.

Since  and , , so . But then  has an angle sum greater than , which is a contradiction. Thus  and must be the same point, so the radius from the center of the circle to the point of tangency is perpendicular to the tangent line, as desired.

Finding the Tangent Line at a Point

Given a circle and a point on the circle, it is relatively easy to find the tangent line using coordinate geometry. For example,