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Unformatted text preview: 72 CHAPTER 1. COORDINATES AND VECTORS A famous formula for the area of a triangle in terms of the lengths of its
sides is the second of two area formulas proved by Heron of Alexandria (ca.
75 AD) in his Metrica :
A= s(s − a)(s − b)(s − c) (1.25) where a, b and c are the lengths of the sides of the triangle, and s is the
s = (a + b + c).
Equation (1.27) is known as Heron’s formula, although it now seems
clear from Arabic commentaries that it was already known to Archimedes
of Syracuse (ca. 287-212 BC). In Exercise 15 and Exercise 16 we will
consider both of the area formulas given in the Metrica ; also, in Exercise 5
we will derive a vector formula for the area of a triangle based on the
discussion of the distance from a point to a line on p. 52.
Here, however, we will concentrate on ﬁnding a formula for the area of a
triangle in R2 in terms of the coordinates of its vertices. Suppose the
vertices are A(a1 , a2 ), B (b1 , b2 ), and C (c1 , c2 ). Using the side AB as the
base, we have
b = AB
and, letting θ be the angle at vertex A,
h = AC sin θ,
AB AC sin θ.
To express this in terms of the coordinates of the vertices, note that
A (△ABC ) = −
AB = xAB − + yAB −
xAB = b1 − a1 yAB = b2 − a2 and similarly
AC = xAC − + yAC − .
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