Proposition

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 = d₁, AM = m, and CN = n. Prove that  (1) triangle MBN is isosceles: d = d₁;  (2) triangles AMB , BNC and ABC are similar;  (3) a² = b.n, similarly c² = b.m;  (4) d² = m.n;  (5) a² + c² = b.(m+n) Qurra's theorem;  (6) 1/a² + 1/c² = (m+n)/(b.m.n).
 

See solution below

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Geometric Art: Hyperbolic Kaleidoscope of problem 1386

See also
Conformal Mapping or Transformation of Problem 1386