স্পর্শক ও লম্ব ব্যাসার্ধ

A tangent is a line which only touches a circle at $1$ single point on its circumference. When the tangent touches the circle, the radius touching that point on the circumference produces a right angle with the tangent. There is a diagram below with the crossing point labeled $P$, the center of the circle labeled $C$, and the right angle drawn:

$CP$ is the perpendicular radius, $XP$ is the tangent, and the $\angle{XPC}$ is the right angle.

A tangent to a circle is a line that touches the circle at exactly one point.

An important theorem regarding tangent lines is the following: A line tangent to a circle is perpendicular to the radius drawn to the point of tangency.

Why is this true? To prove it, we will need three other key facts:

(1) The perpendicular segment is the shortest segment from a point to a line. (easily proved using the Pythagorean Theorem)

(2) The interior of a circle is the set of all points such that the distance to the center is less than the radius. The exterior of a circle is the set of all points such that the distance to the center is greater than the radius. (fairly obvious)

(3) A line passing through the interior of a circle must intersect the circle twice. (also fairly obvious)

Suppose that line $l$ is tangent to a circle with center $O$, and that the point of tangency is $P$. We prove the theorem using contradiction: suppose $OP$ is not perpendicular to $L$. Then, drop a perpendicular from $O$ to $l$ and call the foot of the altitude $P'$. Then, by statement (2), $OP' < OP$. However, by statement (1), $P'$ must lie in the interior of the circle. This would imply that the tangent line passes inside the circle and thus, by statement (3), must intersect the circle twice, a contradiction. It follows that $OP$ must be perpendicular to $L$.