# Vecten's Theorem

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

Lines AA

_{b}and BB_{a}meet on the C-altitude of ΔABC.Lines AA

_{b}and CC_{b}are perpendicular and equal. (This follows from the congruence of triangles ABA_{b}and C_{b}BC.) Assume AA_{b}and CC_{b}meet at point S_{b}and introduce similarly S_{a}and S_{c}.Line A

_{c}C_{a}passes through S_{b}and bisects a pair of angles at that point. (Quadrilateral AC_{a}C_{b}S_{b}is cyclic as having opposite angles at C_{a}and S_{b}both 90°. In the circumcircle, angles AS_{b}C_{a}and AC_{b}C_{a}, the latter being 45°, are subtended by the same chord AC_{a}. Hence angle AS_{b}C_{a}is also 45°.)Lines AS

_{a}, BS_{b}, and CS_{c}are concurrent and each passes through the center of one of*Vecten's squares*.Let T

_{a}, T_{b}, and T_{c}denote the centers of Vecten's squares BCA_{c}A_{b}, etc. Then AT_{a}is equal and perpendicular to T_{b}T_{c}.A

_{c}B_{c}^{2}+ B_{a}C_{a}^{2}+ A_{b}C_{b}^{2}= 3(AB^{2}+ BC^{2}+ AC^{2}).

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