200 HISTORY OF THE THEORY OF NUMBERS. [CHAP. V 



Its area is rational if (a + 3d) (a d) = 3y 2 . Hence decompose 3?/ 2 in 

 every way into two factors congruent modulo 4. 



E. N. Barisien 50 noted that, if 2p is the perimeter, the area is an integer if 



p = (an + l)(|Sw + 1), p -b = \n(jn + 1), 

 p a = (an + l)(yn + 1), p - c = jm(/3n + 1), 



and X/z = ft 2 . The condition p = S(p a) is satisfied if 4y + = 5a, 

 _ <y = 5. If in p b and p c we replace Xn and jun by dn + 1, the 

 area is integral and the condition p = S(p a) gives for 5n + 1 a value 

 which is integral if /3 + 7 = 2; then B is a right angle. A. Gerardin noted 

 that we may set p = (an + t)(pn + 0> etc., and take + 7 = 2a, 

 j8 7 = 2p, = (p 5)?i. 



L. Aubry 51 noted that the triangle with the sides x 1, x, x + 1 has an 

 integral area if (z/2) 2 3y 2 = 1, i. e., if 



x = 2, 4, 14, -, z n = 4z n _i - z n _ 2 . 



The area 52 of any triangle with integral sides and area is a multiple of 6. 



B. Hecht 53 discussed triangles whose sides are integers, also the area 

 or the four radii of the escribed and inscribed circles. 



A. Martin 54 proved that in any primitive rational triangle two sides are 

 odd, the least side is > 2, the difference between the sum of the two smaller 

 sides and the largest side is not unity, and the area is a multiple of 6. 

 Every integer > 2 is the least side of an infinitude of primitive rational 

 triangles. 



E. N. Barisien 55 noted that the triangle with the sides 7, 15, 20 has its 

 area and perimeter each 42. Multiplying the sides by 10, we get a triangle 

 with integral altitudes. 



* H. Bottcher 56 gave rational triangles with an angle 60 or 120. 



Barisien 57 gave complicated formulas for the integral sides of a triangle, 

 with integral values for the altitudes, area, radius of circumscribed circle, 

 radii of tritangent circles, segments of the sides made by the altitudes, and 

 segments of the altitudes made by the orthocenter. 



Of several triangles 58 with integral sides, area and one altitude, the 

 least appears to have the sides 4, 13, 14, area 24 and altitude (to side 4) 12. 



A. Martin 59 added 61 rational scalene triangles to Heppel's 81 list. 



N. Gennimafcas 60 proved that any rational triangle is similar to one with 



60 Sphinx-Oedipe, 5, 1910, 57-9. 

 *Ibid., 6, 1911, 188. 



62 Math. Quest. Educ. Times, 21, 1912, 17-8. See paper 18 above. 



63 Ueber rationale Dreiecke, Wiss. Beil. z. Jahresber. Stadt Realschule in Konigsberg, 1912, 7 pp. 



64 School Science and Math., 13, 1913, 323-6. 



65 Mathesis, (4), 3, 1913, 14, 67. 



M Unterrichtsblatter fur Math. u. Naturwiss., 19, 1913, 132-3. 



67 Sphinx-Oedipe, 8, 1913, 182-3; 9, 1914, 74-5, 91, 94. Assoc. frang. av. sc., 43, 1914, 48-57. 



Mathesis, (4), 4, 1914, 114-6 for 7 examples. 



68 L'interme"diaire des math., 21, 1914, 76, 143, 186-8; 22, 1915, 119-120. 



69 Math. Quest. Educ. Times, 25, 1914, 76-8. 

 80 L'enseignement math., 16, 1914, 48-53. 



