# 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 |