Task. Using the compass and straightedge, construct to a given circle the tangent lines through a given point outside the circle.

Solution. Let $O$ be the centre of the given circle and $P$ the given point. With $OP$ as diameter, draw the circle (see midpoint). If $A$ and $B$ are the points where this circle intersects the given circle, then by Thales' theorem, the angles $OAP$ and $OBP$ are right angles. According to the definition of the tangent of circle, the lines $AP$ and $BP$ are required tangents.

The line segment $AB$ is

The convex angle $APB$ is called a tangent angle (or tangent-tangent angle) of the given circle and the convex angle $AOB$ the corresponding central angle. It is apparent that a tangent angle and the corresponding central angle are supplementary. The chord $AB$ is the tangent chord corresponding the tangent angle and the point $P$ (see equation of tangent chord!).

The tangent angle is the angle of view of the line segment $AB$ from the point $P$ .

Note that if a circle is inscribed in a polygon, then the angles of the polygon are tangent angles of the circle and the centre of the circle is the common intersection point of the angle bisectors.