. Carnegie Institution of Washington publication. CHAPTER V. TRIANGLES, QUADRILATERALS AND TETRAHEDRA WITH RATIONAL SIDES. RATIONAL OR HERON TRIANGLES. Heron of Alexandria gave the well known formula for the area of a triangle in terms of the sides and noted that when the sides are 13, 14, 15, the area is 84. A triangle with rational sides and rational area is called a rational triangle or Heron triangle. Brahmegupta1 (born 598 ) noted that, if a, b, c are any rational numbers, are sides of an oblique triangle [whose2 altitudes and area are rational and which is formed by the juxtaposition
. Carnegie Institution of Washington publication. CHAPTER V. TRIANGLES, QUADRILATERALS AND TETRAHEDRA WITH RATIONAL SIDES. RATIONAL OR HERON TRIANGLES. Heron of Alexandria gave the well known formula for the area of a triangle in terms of the sides and noted that when the sides are 13, 14, 15, the area is 84. A triangle with rational sides and rational area is called a rational triangle or Heron triangle. Brahmegupta1 (born 598 ) noted that, if a, b, c are any rational numbers, are sides of an oblique triangle [whose2 altitudes and area are rational and which is formed by the juxtaposition of two right triangles with the common leg a]. S. Curtius20 proposed the following question: Three archers A, B, and C stand at the same distance from a parrot, B being 66 feet from C, B 50 feet from A, and A 104 feet from C; if the parrot rises 156 feet from the ground, how far must the archers shoot to reach the parrot? He noted that they stand at the vertices of a triangle the radius of whose circum- scribed circle is 65 feet, while the parrot is 156 feet above its center. Since 652 + 1562 = 1692, each archer is 169 feet from the parrot. It is stated to be difficult to explain why the radius turns out to be an integer. Cf. [The triangle is rational since its area is 23 • 3 • 5 • 11 = 1320.] C. G. Bachet,3 in his comments on Diophantus VI, 18, treated several problems, the second of which is to find a triangle with rational sides and a rational altitude (and hence a Heron triangle). Taking a right triangle ADC with the sides 10, 8, 6, he found BD = N such that N2 + 82 shall be the square of a rational number (AB). Assuming first that angle B A C is acute, so that DC : AD < AD : BD, we must have 6N < 64, whence N < 32/3. Let N2 + 82 be the square of 8 - xN; then. 10 1Qx - N 32 < 3 ' x2 — 1 •^ -*- •» r ^ / — /V ^ — ( 1 '2 _. 1 i X'Y* 1), OX + 2 < 2z2, x* -• I 1C •i* ^ 0 ^ 16 o 1 Brahme-Sphut'a-Sidd'hanta, Ch. 12, Sec. 4, § 34 Alge
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