CHAP. V] TRIANGLES WHOSE AREA NEED NOT BE RATIONAL. 215 



harmonical progression. Its sides are the halves of the sides of a triangle 

 whose area is divisible by each side. A. B. Evans 118 noted that the altitudes 

 Pi vary inversely as the sides a, b, c, whence the condition is a + c = 26. 

 Let x = cot %A, y = cot \B. Thus 2y = x + (x + y)/(xy^ 1), which 

 gives y rationally. Then, if r is the radius of the inscribed circle, 



a = r (cot %B + cot |C), , Pi = r(a + & + c)/a, . 



Evans and A. Martin 119 found rational triangles with integral sides and 

 lines from the vertices to the center of the inscribed circle, by use of 

 OA = r esc ^A. 



M. Weill stated and E. Cesaro 120 proved that (4, 5, 6) is the only triangle 

 whose sides are consecutive integers and the ratio of two of whose angles 

 is an integer. 



K. Schwering 121 noted that the ratios of the sines of the three angles 

 a, j8, 7 are rational if the sides are rational. Assigning values to tan a/2 

 and tan /3/2, whose ratio is rational, we have tan y/2 and hence the ratios 

 of a b c and therefore the ratios of a, 6, c. He discussed the problem 

 to find a point inside an equilateral triangle with the given rational side a 

 such that the distances from to the vertices shall be rational. 



Ziige 122 gave the general solution of z 2 = x 2 + y z 2xy cos a, where 

 cos a. is rational. [But the topic is of little interest since we obtain a 

 triangle with rational sides x, y, z by assigning to them any rational values 

 such that x + y > z, etc.] 



A. B. Evans 123 noted that, if BC = 399, AC = 455, AB = 511, CO = 195, 

 BO = 264, AO = 325, the lines joining to the vertices of triangle ABC 

 make equal angles. 113 



Several 124 gave triangles with integral sides and an angle 60. 



A. Martin 125 discussed the last problem. 



R. A. Johnson 126 gave expressions for the integral sides of any triangle 

 with a given rational value for the cosine of one angle. 



Several 127 gave pairs of triangles with integral sides having a common 

 base and equal altitudes. 



E. Turriere 128 found points whose distances from the three vertices of 

 a given triangle with rational sides are all rational. 



N. Alliston 129 gave special triangles with integral sides and points whose 

 distances from the vertices are integers. 



118 Math. Quest. Educ. Times, 22, 1875, 54. 



119 Ibid., 102-3. 



120 Mathesis, 9, 1889, 142-3. Also proof by Weill, Nouv. Ann. Math., (4), 14, 1914, 526-7. 



121 Geom. Aufgaben mit rationalen Losungen, Progr. Diiren, 1898. 

 122 Archiv Math. Phys., (2), 17, 1900, 354. 



123 Math. Quest. Educ. Times, 72, 1900, 77. 



124 Zeitschrift Math. Naturw. Unterricht, 45, 1914, 184-5. 



1 25 Amer. Math. Monthly, 21, 1914, 98-9. Cf. Neuberg 3 ". of Ch. XIII. 

 Ibid., 22, 1915,27-30. 



127 Math. Quest. Educ. Times, 27, 1915, 91-2. 



128 L'enseignement math., 19, 1917, 262-7. 



129 Math. Quest, and Solutions, 5, 1918, 37. 



