A tangent to a circle is a line intersecting the circle at exactly one point, the point of tangency or tangency point. An important result is that the radius from the center of the circle to the point of tangency is perpendicular to the tangent line.
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.
Given a circle and a point on the circle, it is relatively easy to find the tangent line using coordinate geometry. For example,
A circle of radius is centered at the origin. What is the equation of the tangent line of the circle that passes through the point
The slope of the line from the center of the circle (the origin) to the given point is . Therefore, the slope of the tangent line is 1, and since the tangent line passes through the point the equation of the tangent line is
The perpendicularity fact can also prove a special (but important!) case of power of a point. Suppose is a point in the plane of a circle , so that is tangent to the circle at and hits the circle at and with Then is right, so
which is exactly what power of a point predicts since .
The perpendicularity condition is particularly useful when dealing with multiple circles, as their common tangent must be perpendicular to both radii to the tangent points. This also implies that those two radii are parallel, so the tangent line, two radii, and the line between the two centers form a trapezoid.
Two externally tangent circles and have radii of 3 and 5, respectively. What is the length of their common external tangent?
Suppose the common tangent is tangent to the circle with radius 3 at and to the circle with radius 5 at . Let be the foot of the altitude from to . Then , and . But , and , so . Therefore, , which is the desired length.