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 the location of "P". Pompeiu's theorem states that, if "P" is an arbitrary point in an equilateral triangle "ABC", then there exists a triangle with sides of lengths "PA", "PB", and "PC". That is, "PA", "PB", and "PC" satisfy the triangle inequality that any two of them sum to at least as great as the third. By Euler's inequality, the equilateral triangle has the smallest ratio "R"/"r" of the circumradius to the inradius of any triangle: specifically, "R"/"r" = 2. The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest