A tangent is a straight line that touches a curve at only one point and does not intersect it at that point.


- A circle is a set of points in a plane that are equidistant from a fixed point called the center.
- The distance from the center to any point on the circle is called the radius.
- All radii of a circle are equal in length.
- The diameter of a circle is twice the length of the radius and passes through the center of the circle.
- The circumference of a circle is the distance around the circle and is equal to 2π times the radius.


Consider a circle with center O and radius r. Let P be a point on the circumference of the circle, and let AB and CD be two distinct lines passing through point P.


In this case, line AB intersects the circle at two distinct points, say Q and R. Since Q and R are points on the circle, their distances from the center O are both equal to r. Thus, the line segment QR is a chord of the circle. By the properties of a circle, all chords passing through a given point on the circumference of a circle are congruent. Since PQ and PR are both radii of the circle, they are congruent. Therefore, triangle PQR is an isosceles triangle, and the altitude from P to QR bisects QR, dividing it into two congruent segments. But since AB intersects QR at point S, the altitude from P to QR must also be perpendicular to AB. Therefore, AB and QR cannot be distinct lines passing through point P, which contradicts our assumption. Hence, AB must be tangent to the circle at point P.


This case can be proven similarly to Case 1.


In this case, AB and CD are both perpendicular to the radius OP at point P. By the properties of a circle, any two lines that are perpendicular to the same radius of a circle must be parallel. Therefore, AB and CD are parallel, which contradicts our assumption that they are distinct lines passing through point P. Hence, this case is not possible.

Therefore, we have proven that there is at most one tangent to a circle at any point on its circumference. Since a tangent must exist at any point on the circumference of a circle, there is exactly one tangent at any point on the circle.