Vecten's Theorem

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

1. Lines AA_b and BB_a meet on the C-altitude of ΔABC.

2. Lines AA_b and CC_b are perpendicular and equal. (This follows from the congruence of triangles ABA_b and C_bBC.) Assume AA_b and CC_b meet at point S_b and introduce similarly S_a and S_c.

3. Line A_cC_a passes through S_b and bisects a pair of angles at that point. (Quadrilateral AC_aC_bS_b is cyclic as having opposite angles at C_a and S_b both 90°. In the circumcircle, angles AS_bC_a and AC_bC_a, the latter being 45°, are subtended by the same chord AC_a. Hence angle AS_bC_a is also 45°.)

4. Lines AS_a, BS_b, and CS_c are concurrent and each passes through the center of one of Vecten's squares.

5. Let T_a, T_b, and T_c denote the centers of Vecten's squares BCA_cA_b, etc. Then AT_a is equal and perpendicular to T_bT_c.

6. A_cB_c² + B_aC_a² + A_bC_b² = 3(AB² + BC² + AC²).

Explanation

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