In triangle ABC, let E be the midpoint of altitude AD from A to BC. The excircle opposite C is tangent to BC at G, and line EG intersects the excircle at H. Prove that HG bisects angle BHC.

In triangle's grace,

Midpoint, altitude, and excircle,

Bisectors reveal.

Key Term | Description |
---|---|

Triangle | The geometric figure ABC, consisting of three sides and three angles. |

Altitude | The perpendicular segment AD from vertex A to side BC in triangle ABC. |

Midpoint | The point E that divides the altitude AD into two equal segments. |

Excircle | The circle opposite vertex C that is tangent to side BC of triangle ABC at point G. |

Tangency Point | Point G where the excircle opposite C is tangent to BC. |

Perpendicular | The line segment AD, which is perpendicular to side BC of triangle ABC. |

Intersection | The point H where line EG intersects the excircle opposite C. |

Angle Bisector | The line HG that bisects angle BHC in the geometric problem. |

Bisects Angle | To divide an angle into two equal parts. In this problem, line HG bisects angle BHC. |

Geometry Problems

Open Problems

Visual Index

All Problems

Triangles

Altitude

Midpoint

Circle

Triangle Centers

Excircle

Secant to a Circle

Circle Tangent Line

Angle

Angle Bisector

Perpendicular lines

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