A history of Greek mathematics - Wilbourhall.org

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AREA OF SCALANE TRIANGLE 321
from the opposite vertex, and thence the length of the perpendicular
itself. We have, in the cases of the triangle acuteangled
at C and the triangle obtuse-angled at C respectively,
c = 2 a 2 + b 2 +2a.CD,
or
GD = {(«2 + 2 /; )-c 2 }/2^,
whence AD 2 (= b 2 — GD 2 ) is found, so that we know the area
(=ia.AD).
In the cases given in Metrica I. 5, 6 the sides are (14, 15, 13)
and (11, 13, 20) respectively, and AD is found to be rational
(=12). But of course both CD (or BD) and AD may be surds,
in which case Heron gives approximate values. Cf. Geom.
53, 54, Hultsch (15, 1-4, Heib.), where we have a triangle
in which a = 8, 6=4, c = 6, so that a 2 + b 2 2
— c = 44 and
CD = 44/16 = 2|i. 2
Thus AD'= 16-(2|J) = 16-7| TV
= 8 i ¥ Te and AD= ' -/(8J | ye) = 2§ i approximately, whence
the area = 4 x 2§ J = 11§. Heron then observes that we get
a nearer result still if we multiply AD 2 2
by (J a) before
extracting the square root, for the area is then