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
Triangle (in fact, this formula holds for "all" tangential polygons), and where formula_59 are the radii of the excircles tangent to sides "a, b, c" respectively. We also have and for circumdiameter "D"; and for angle α ≠ 90°. The area can also be expressed as In 1885, Baker gave a collection of over a hundred distinct area formulas for the triangle. These include: for circumradius (radius of the circumcircle) "R", and The area "T" of any triangle with perimeter "p" satisfies with equality holding if and only if the triangle is equilateral. Other upper bounds on the area "T" are