Vecten's Theorem


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Following is a listing of a few properties of Vecten's configuration:

  1. Lines AAb and BBa meet on the C-altitude of ΔABC.

  2. Lines AAb and CCb are perpendicular and equal. (This follows from the congruence of triangles ABAb and CbBC.) Assume AAb and CCb meet at point Sb and introduce similarly Sa and Sc.

  3. Line AcCa passes through Sb and bisects a pair of angles at that point. (Quadrilateral ACaCbSb is cyclic as having opposite angles at Ca and Sb both 90°. In the circumcircle, angles ASbCa and ACbCa, the latter being 45°, are subtended by the same chord ACa. Hence angle ASbCa is also 45°.)

  4. Lines ASa, BSb, and CSc are concurrent and each passes through the center of one of Vecten's squares.

  5. Let Ta, Tb, and Tc denote the centers of Vecten's squares BCAcAb, etc. Then ATa is equal and perpendicular to TbTc.

  6. AcBc2 + BaCa2 + AbCb2 = 3(AB2 + BC2 + AC2).

Explanation

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