652 - PHILOSOPHICAL TRANSACTIONS. [aNNO 1775. 



angles, must be homologous ; but be is greater than bc, which is equal to ab, 

 and that again greater than ad ; consequently be is greater than ad, and the 

 whole triangle aeb greater than aed ; and so the equilateral triangle must, 

 k fortiori, be greater than aed. q.e.d. 



Next, let the triangle ade be supposed greater than the equilateral triangle 

 ABC ; and let the angle ade fall somewhere in the segment bdc, (fig. l6,) so 

 that the segment ec may be greater than bd ; for if it were not, the angle aed 

 being applied to any of the angles of the equilateral triangle, the demonstration 

 would become the same as in the first case : therefore the segments aec, bdc, 

 being equal, and bd being less than ec, ae must be less than dc. Draw the 

 right line dc ; then, in the two triangles adc, ade, the triangle afd is common, 

 and the two triangles afe, dpc are equiangular and similar, and the sides ae, dc, 

 subtending equal angles, are homologous ; but dc is greater than ae ; so the 

 triangle dfc is greater than the triangle afe, and the whole triangle adc is 

 greater than ade ; but the equilateral triangle may be proved greater than adc 

 from the first case, and consequently greater than ade. a.E.o. 



Prop. 2. ^n equilateral triangle described about a circle is less than any other 

 triangle that can be described about the same circle. — Let the equilateral triangle 

 ABC, fig. 17, be described about the circle hik, and let the triangle bdg be 

 supposed less than the equilateral triangle. Draw the line af parallel to bc ; 

 then the triangles AFE, egc, are similar; for the opposite angles aef, gec, are 

 equal, as likewise the angle afe to the angle egc ; the lines af and gc being 

 parallel, and falling on the same line fg, the angles afe and egc are therefore 

 equal, and the sides ae, ec, subtending equal angles, are homologous ; but the 

 side of the equilateral triangle ac being equally divided at i, the line ae is greater 

 than EC, and consequently the triangle afe is larger than the triangle egc ; and 

 the triangle dae much larger than egc : therefore, in the triangles DBGandABC, 

 the part abge being common, the whole triangle dbg is larger than the equi- 

 lateral triangle. a.E.D. 



Whatever other triangles can be described about a circle, may be demonstrated 

 to be larger than an equilateral triangle described about the same circle, on the 

 same principles as the preceding. 



Prop. 3. The square of the side of an equilateral triangle, inscribed in a 

 circle^ is equal to a rectangle under the diameter of the circle, and a perpen- 

 dicular let fall from any angle of the triangle on the opposite side. — The two 

 triangles ADC, aec, fig. 18, are equiangular and similar, the angles acd, aec, 

 being both right, and that at a common ; therefore ad : ac :: ac : ae, and 



AC'* = AD X AE. Q.E.D. 



The square of one side of the triangle being compleated, so as to include the 

 triangle ; then that part of the side of the square that falls within the circle is 



