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
Malfatti circles corners of the triangle, the one with the smallest angle, and inscribes a third circle within the largest of the five remaining pieces. The difference in area for an equilateral triangle is small, just over 1%, but as pointed out, for an isosceles triangle with a very sharp apex, the optimal circles (stacked one atop each other above the base of the triangle) have nearly twice the area of the Malfatti circles. The problem of constructing three circles tangent to each other within a triangle was posed by the 18th-century Japanese mathematician Ajima Naonobu prior to the work of Malfatti,