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
Isodynamic point point is the intersection of three circles that intersect the circumcircle to form lenses with apex angle π/3. The angles formed by the first isodynamic point with the triangle vertices satisfy the equations formula_29, formula_30, and formula_31. Analogously, the angles formed by the second isodynamic point satisfy the equationsformula_32, formula_33, and formula_34. The pedal triangle of an isodynamic point, the triangle formed by dropping perpendiculars from formula_1 to each of the three sides of triangle formula_3, is equilateral, as is the triangle formed by reflecting formula_1 across each side of the triangle. Among all the equilateral triangles inscribed in triangle