Weber problem that, at the optimum, the angles between the AD, BD, and CD straight lines must be equal to 360° / 3 = 120°. Torricelli deduced from that conclusion that: 1– if any triangle ABD, whose ∠ADB angle is equal to 120°, generates an ABDE convex quadrilateral inscribed in a circle, the ∠ABE angle of the ABE triangle must be equal to (180° − 120°)= 60°; 2– one way to determine the set of locations of D for which the ∠ADB angle is equal to 120° is to draw an equilateral ABE triangle (because each angle of an equilateral triangle is
Automedian triangle sharing no common factor) as with formula_4 and formula_5 coprime, formula_6 odd, and to satisfy the triangle inequality formula_7 (if the quantity inside the absolute value signs is negative) or formula_8 (if that quantity is positive). Then this triangle’s medians formula_9 are found by using the above expressions for its sides in the general formula for medians: where the second equation in each case reflects the automedian feature formula_11 From this can be seen the similarity relationships There is a primitive integer-sided automedian triangle that is not generated from a right triangle: namely, the equilateral triangle with sides of unit