Herons formula: Approximately 2000 years ago, Heron of

Chapter 9, Problem 60

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Herons formula: Approximately 2000 years ago, Heron of Alexandria derived a formula for the area of a triangle in terms of the lengths of the sides. A more modern derivation of Herons formula is indicated in the steps that follow. (a) Use the expression for sin A in Exercise 56(b) to show that Hint: Use difference-of-squares factoring repeatedly. (b) Let s denote one-half of the perimeter of ^ABC. That is, let s (a b c). Using this notation (which is due to Euler), verify that (i) a b c 2s (ii) a b c 2(s a) (iii) a b c 2(s b) (iv) a b c 2(s c) Then, using this notation and the result in part (a), show that Note: Since sin A is positive (Why?), the positive root is appropriate here. (c) Use the result in part (b) and the formula area ^ABC sin A to conclude that This is Herons formula. For historical background and a purely geometric proof, see An Introduction to the History of Mathematics, 6th ed., by Howard Eves (Philadelphia: Saunders College Publishing, 1990), pp. 178 and 194. area ^A