In a right-angled triangle ABC, where angle ABC is 90 degrees, let BH be the altitude from B to AC and BD be the bisector of angle ABC. The bisector of angle AHB intersects AB at E, and the bisector of angle BHC intersects BC at F. If M is the midpoint of BD, prove that the points E, M, and F are collinear.

Right triangle stands tall,

Bisectors meet, paths align,

E, M, F in line.

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

Right-Angled Triangle ABC | A triangle with one angle measuring 90 degrees, with vertices labeled as A, B, and C. |

Angle ABC | The right angle (90 degrees) in the triangle, located at vertex B. |

Altitude BH | A perpendicular line segment from vertex B to the hypotenuse AC. |

Bisector BD | A line segment from vertex B that bisects angle ABC, meeting AC at point D. |

Point E | The point where the bisector of angle AHB intersects side AB. |

Point F | The point where the bisector of angle BHC intersects side BC. |

Midpoint M | The point that is exactly halfway along BD. |

Collinear Points E, M, F | The points E, M, and F lie on a single straight line. |

Geometry Problems

Open Problems

Visual Index

All Problems

Angles

Triangle

Right Triangle

Angle Bisector

Perpendicular lines

Midpoint

Altitude

Collinear Points

Angle
of 45 degrees

View or Post a solution