Euler's Triangle Theorem
نویسنده
چکیده
Theorem: Let A; B; C be an arbitrary triangle and O any point of the plane which does not lie on a side of the triangle. Let AO, BO,COmeet BC,CA and AB in the points D, E, F respectively. Then jjAOjj jjODjj + jjBOjj jjOEjj + jjCOjj jjOFjj = jjAOjj jjODjj jjBOjj jjOEjj jjCOjj jjOFjj + 2. 1 B D C A E F O Figure 1 T o begin with we shall assume, like Euler, that O lies in the interior of the triangle, see Figure 1. For the present, the notation jjXYjj jjY Z jj will simply indicate the ratio of the lengths of the indicated segments. We describe this theorem as remarkable not only because of its early date, but also because it connects the value of a cyclic sum on the left side of 1 with the value of a cyclic product on the right side of 1. There e xist a great number of results c oncerning cyclic products for triangles and ngons, the best known of which are the theorems of Ceva and Menelaus. Many more such theorems are given in 66 and 77. On the other hand there a r e few papers o n c y clic sums, such as 55 and 88. Apart from 11, Euler's paper 44 is, so far as we are aware, the only one which related these two concepts. Euler's proof of his theorem is by algebra and trigonometry; he calculates the ratios of the lengths of the line segments in 1 using trigonometrical formulae involving the sines of the angles between the lines at O. The proof takes two and a half pages, and whilst we do not wish to criticize the work of one of the most illustrious of mathematicians, it is worth considering the shortcomings of his proof. Apart from its length and complexity, the proof gives no insight as to why relation 1 is true, and therefore does nothing to
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