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Pairs of Related Triangles Studied by Complex Coordinates

Published online by Cambridge University Press:  03 November 2016

Extract

Let α1, β1, γ1, and α2, β2, γ2 be the complex numbers corresponding to the angular points A1, B1, C1 and A2, B2, C2 of two related triangles, T1, T2, in a plane, and let homologous vertices be denoted by the same letters.

Type
Research Article
Copyright
Copyright © The Mathematical Association 1953

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References

page no 1 note * The number δαα remains unaltered only when the orientation of the coordinate axes is preserved, one change of sense in this transforming δαα to δαα The invariant depends, therefore, on the two triangles and the positive sense of rotation adopted in the plane.

page no 2 note * We express the fact that a number z, which is in general complex, has in a particular instance a purely real value, without specifying that value, by writing z = [r]. Similarly, the fact that z is purely imaginary is denoted by writing z =[pi].

page no 2 note † The pairs of homologous vertices are, besides, the same in the two correspondences in which the triangles are orthologic and metaparallel.

page no 4 note * See Agronomof, , Revista matemática hispano-americana, 1927, pp. 299304.Google Scholar

page no 5 note * |φ| = 1 implies the existence of one point common to the three circles, but as a consequence of this, the circles meet in a second point. For, if P is the first point common to the three circles, there exists another point Q through which the circles pass, since the equations

are certainly compatible. P and Q are, besides, the two points in the plane whose distances from the vertices A l, B 1, C 1 are proportional to three given lengths.