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 diameterMathworldPlanetmathPlanetmath, draw the circle (see midpointMathworldPlanetmathPlanetmathPlanetmath). 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 anglesMathworldPlanetmathPlanetmath. According to the definition of the tangent of circle, the lines AP and BP are required tangents.

The line segmentMathworldPlanetmath 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 angleMathworldPlanetmath. 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 inscribedMathworldPlanetmath in a polygonMathworldPlanetmathPlanetmath, 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 bisectorsMathworldPlanetmath.

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 IncircleMathworldPlanetmath
Related topic AngleBisectorAsLocus
Defines tangent angle
Defines tangent-tangent angle
Defines tangent chord