AB = a, BC = b, CD = c, DA = d, ZDAB = x, ZBCD = y; see Figure 1. A proof that uses a different approach is given by A. Varverakis in. 9. [3]. In this note, we give a ...
Appeared as Note 100.22 in Math. Gaz. 100 (July 2016), 335–336
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The maximal area property of cyclic quadrilaterals It is well known that among all quadrilaterals with given side lengths, the cyclic quadrilaterals have maximal area. Proofs of this can be found, for example, in [1, p. 7] and [2, p. 49], and these proofs use Brahmagupta’s formula 16K 2 = 4(ad + bc)2 − (a2 + d2 − b2 − c2 )2 − 8abcd cos(x + y)
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that gives the area K of a convex quadrilateral ABCD whose side lengths and angles are given by AB = a, BC = b, CD = c, DA = d, ∠DAB = x, ∠BCD = y;
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see Figure 1. A proof that uses a different approach is given by A. Varverakis in [3]. In this note, we give a short proof using Lagrange’s multipliers. Neither the existing proofs nor this one is Euclidean, in the sense that Euclid would enjoy reading. C y
b B
c u
a x
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Figure 1
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A
Referring to Figure 1, we use the law of cosines to obtain u2 = a2 + d2 − 2ad cos x and u2 = b2 + c2 − 2bc cos y. Subtracting, we obtain f (x, y) = (a2 + d2 − 2ad cos x) − (b2 + c2 − 2bc cos y) = 0.
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The area of ABCD is given by 1 (ad sin x + bc sin y) . 2 Thus we are to maximize g(x, y) given the constraint f (x, y) = 0. According to the method of Lagrange, we are to solve the system [∇g = λ∇f, f (x, y) = 0], i.e., the system g(x, y) =
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(i)
ad cos x bc cos y = λ(2ad sin x), (ii) = λ(−2bc sin y), (iii) f = 0. 2 2 1
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If λ = 0, then cos x = cos y = 0, and hence x = y = π/2 and the quadrilateral is cyclic. This actually happens, in view of (iii), when a2 + d2 = b2 + c2 . So we assume that λ ̸= 0. In this case, cos x ̸= 0; otherwise, sin x = ±1 and (i) implies that λ = 0. Therefore cos x ̸= 0, and similarly cos y ̸= 0. It now follows from (i) and (ii) that tan x = − tan y, each being equal to 1/(4λ). Since x, y > 0 and x + y < 2π, it follows that x + y = π, and hence the optimal quadrilateral is cyclic, as claimed.
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References
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[1] Z. A. Melzak, Invitation to Geometry, John Wiley & Sons, New York, 1983. [2] N. D. Kazarinoff, Geometric Inequalities, New Mathematical Library, No. 4, Math. Assoc. America, Washington D.C., 1961. [3] A. Varverakis, A maximal property of cyclic quadrilaterals, Forum Geom. 5 (2005), 63–64.
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