The figure below shows a triangle ABC of sides a, b, c
with the cevians BM and BN so that the angles A and C are congruent to the
angles NBC and ABM, respectively. BM = d, BN = d_{1}, AM = m, and CN = n.
Prove that (1) triangle MBN is isosceles: d = d_{1}; (2) triangles AMB , BNC and ABC are similar; (3) a^{2}
= b.n, similarly c^{2} = b.m; (4) d^{2} = m.n; (5) a^{2} + c^{2} = b.(m+n) Qurra's
theorem; (6) 1/a^{2} + 1/c^{2}
= (m+n)/(b.m.n).

Key Term / Hints | Description |
---|---|

Thabit ibn Qurra's Theorem | A generalization of the Pythagorean Theorem to any triangle, involving cevians and angle congruences. |

Triangle ABC | The primary triangle in the problem with sides a, b, and c. |

Cevian | A line segment from a vertex to the opposite side of the triangle, intersecting at a point. |

Isosceles Triangle | A triangle with at least two equal sides. |

Similarity of Triangles | Triangles that have the same shape but not necessarily the same size, meaning their corresponding angles are equal, and their corresponding sides are proportional. |

Generalized Pythagorean Theorem | The relationship among the sides of a triangle and cevians, extended from the classical Pythagorean theorem. |

Proposition Results | Results including the conditions for isosceles triangles, similar triangles, and specific algebraic relationships between sides and cevians. |

See also

Conformal Mapping or Transformation of Problem
1386

Geometry Problems

Ten problems: 1381-1390

Visual Index

Open Problems

All Problems

Triangle

Isosceles Triangle

Similarity, Ratios, Proportions

Right Triangle: Pythagoras

View or Post a solution