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).

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

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