construction of tangent

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

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 (http://planetmath.org/EquationOfTangentOfCircle)!).

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.

Title	construction of tangent
Canonical name	ConstructionOfTangent
Date of creation	2013-03-22 17:36:04
Last modified on	2013-03-22 17:36:04
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Algorithm
Classification	msc 51M15
Classification	msc 51-00
Related topic	Incircle
Related topic	AngleBisectorAsLocus
Defines	tangent angle
Defines	tangent-tangent angle
Defines	tangent chord