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
Equilateral triangle to the points where the angle bisectors of ∠"APB", ∠"BPC", and ∠"CPA" cross the sides ("A", "B", and "C" being the vertices). For any point P in the plane, with distances "p", "q", and "t" from the vertices A, B, and C respectively, For any point P on the inscribed circle of an equilateral triangle, with distances "p", "q", and "t" from the vertices, and For any point P on the minor arc BC of the circumcircle, with distances "p", "q", and "t" from A, B, and C respectively, and moreover, if point D on side BC divides PA into