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# Chapter Five

### Relationships in Triangles

Term | Definition |
---|---|

perpendicular bisector | In a triangle, a line segment, or ray that passes through the midpoint of a side and is perpendicular to that side. |

concurrent lines | three or more lines intersecting at a common point. |

point of concurrency | the point where concurrent lines intersect |

circumcenter | the point of concurrency of the perpendicular bisectors |

incenter | the point of concurrency for the angle bisectors |

Perpendicular Bisector 5.1 Theorem | If a point is on a perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. |

Converse of the Perpendicular Bisector Theorem | If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. |

Circumcenter Theorem | The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle. |

Angle Bisector Theorem | If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. |

Converse of the Angle Bisector Theorem | If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. |

Incenter Theorem | The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from each side of the triangle. |

median | a triangle of a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side. |

altitude | a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. |

centroid | the point of concurrency of the medians of a triangle |

orthocenter | the lines containing the altitudes of a triangle are concurrent and intersect at the orthocenter. |

Angle Side Relationships in Triangle | If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. |

Angle Side Relationship 5.10 | If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. |

Triangle Inequality Theorem | The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. |

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amgeometry