CHAP. V] RATIONAL TRIANGLES. 199 



determines x/y. Taking x to be the numerator of the resulting fraction, 

 we have 



x (p 2 + <? 2 )0" 2 s 2 ), y = 2rs(p z + <? 2 ) =b 2s 2 (p 2 q z ), 



He discussed at length rational triangles two of whose sides differ by a 

 given integer, making use of a Pell equation gq 2 p 2 = 1. 



T. H. Safford 42 juxtaposed the right triangles (5, 12, 13), (9, 12, 15) of 

 areas 30 and 54 to obtain Heron's triangle (13, 14, 15) of area 84, also to 

 obtain (4, 13, 15) of area 54-30. He listed 37 rational right triangles. 



D. N. Lehmer 43 derived (1) by use of the rationality of the sines and 

 cosines of the three angles, a necessary and sufficient condition for the 

 rationality of the triangle. 



Rational triangles with consecutive integral sides have been found. 44 



W. A. Whitworth and D. Biddle 45 proved that there are only five 

 triangles with integral sides whose area equals the perimeter: (5, 12, 13), 

 (6, 8, 10), (6, 25, 29), (7, 15, 20), (9, 10, 17). 



A. Martin 46 formed rational triangles by the juxtaposition of two 

 rational right triangles. He tabulated 168 rational triangles of area ^ 46410 

 not found in Tebay's 19 table. 



H. Schubert 47 considered a Heron triangle with integral sides a, 6, c 

 and area J. If a, j3, y are the angles, / = tan a/2 and hence also sin a 

 and cos a. must be rational (such an angle a being called a Heron angle) . 

 Set / = n/m, where n and m are relatively prime integers. Then 



2mn 2pq . 2(mq + np)(mp nq) 



sin a. = ^. , , sin p = r ; , sin 7 = - ; w ^ , 



m 2 + n 2 p 2 + q z (m 2 + w 2 )(p 2 + g 2 ) 



since tan 7/2 = cot (a + 0)/2. By a = 2r sin a, etc., 



4r = (m 2 + w 2 )(p 2 + <? 2 )- 

 Hence 

 a = mn(p z + g 2 ), b = pq(m 2 + n 2 ), c = (mq + np)(mp nq), J = mnpqc. 



J. Sachs 48 gave tables of rational triangles with altitudes < 100; acute 

 rational triangles with altitudes 100, -, 500; rational triangles arranged 

 according to the least side and according to the greatest side. The last 

 tables are convenient for the formation by juxtaposition of rational quadri- 

 laterals, pentagons, etc. 



T. Harmuth 49 considered rational triangles with sides a, a + d, a + 2cL 



42 Trans. Wisconsin Acad. Sc., 12, 1898-9, 505-8. 



43 Annals of Math., (2), 1, 1899-1900, 97-102. 



44 Amer. Math. Monthly, 10, 1903, 172-3. 



45 Math. Quest. Educ. Times, 5, 1904, 54-6, 62-3. 



46 Math. Magazine, 2, 1904, 275-284. 



47 Die Ganzzahligkeit in der algebraischen Geometric, Leipzig, 1905, 1-16. Festgabe 48 



Versammlung d. Philologen u. Schulmanner zu Hamburg, 1905. Reprinted in Auslese 

 aus meiner Unterrichts- u. Vorlesungspraxis, Leipzig, 2, 1905, 1-23. 



48 Tafeln zum Math. Unterricht, Progr. 794, Baden-Baden, Leipzig, 1908. 



49 Unterrichtsblatter fiir Math. u. Naturwiss., 15, 1909, 105-6. 



