# Relation between the inradius,exradii,circumradius and the distances of the orthocenter from the vertices of a triangle

## Description

Altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the base (the opposite side of the triangle). This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called the altitude, is the distance between the extended base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. Denote the vertices of a triangle as A, B, and C and the orthocenter as H, r as the radius of the triangle’s incircle, ra, rb, and rc as the radii if its excircles, and R as the radius of its circumcircle, then, there is a relation between them.

Related formulas## Variables

r_{a} | Exradius of the tangent excircle to BC side (m) |

r_{b} | Exradius of the tangent excircle to AC side (m) |

r_{c} | Exradius of the tangent excircle to AB side (m) |

r | The inradius (m) |

AH | Distance of the orthocenter from the vertex A (m) |

BH | Distance of the orthocenter from the vertex B (m) |

CH | Distance of the orthocenter from the vertex C (m) |

R | The circumradius (m) |