In triangle ABC, let D be an interior point. Points E, F, and G lie on lines AD, BD, and CD, respectively. The circle through E, D, and F intersects AB at H and I; the circle through D, F, and G intersects BC at J and K; the circle through D, E, and G intersects AC at L and M. Prove that H, I, J, K, L, and M are concyclic.

Three circles converge,

Six points align in a ring,

Geometry's dance.

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

Interior Point D | A point D located inside triangle ABC. |

Points E, F, G | Points lying on lines AD, BD, and CD, respectively. |

Circle through E, D, F | A circle passing through points E, D, and F, intersecting AB at points H and I. |

Circle through D, F, G | A circle passing through points D, F, and G, intersecting BC at points J and K. |

Circle through D, E, G | A circle passing through points D, E, and G, intersecting AC at points L and M. |

Concyclic Points H, I, J, K, L, M | Points H, I, J, K, L, and M lie on a common circle, indicating their concyclicity. |

Geometry Problems

Open Problems

Visual Index

All Problems

Triangles

Circle

Intersecting Circles

Secant to a Circle

Concyclic
Points

Cyclic Quadrilateral

View or Post a solution