By H Dudeney
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Extra info for 536 PUZZLES & CURIOUS PROBLEMS
58 K = B = 25◦ Case 3: Three sides.
The other solution set can be obtained similarly. Like the Law of Sines, the Law of Cosines can be used to prove some geometric facts, as in the following example. 9 Use the Law of Cosines to prove that the sum of the squares of the diagonals of any parallelogram equals the sum of the squares of the sides. 2. Then we need to show that a d C By the Law of Cosines, we know that c2 = a2 + b2 − 2ab cos C , and d 2 = a2 + b2 − 2ab cos D . By properties of parallelograms, we know that D = 180◦ − C , so d 2 = a2 + b2 − 2ab cos (180◦ − C ) = a2 + b2 + 2ab cos C , since cos (180◦ − C ) = − cos C .
1 43 If C is obtuse, then 180◦ − C is acute, as are A and B. If A > 180◦ − C then A + C > 180◦ , which is impossible. Thus, we must have A ≤ 180◦ − C . Likewise, B ≤ 180◦ − C . So by what we showed above for acute angles, we know that sin A ≤ sin (180◦ − C ) and sin B ≤ sin (180◦ − C ). 5 that sin C = sin (180◦ − C ). Hence, sin A ≤ sin C and sin B ≤ sin C when C is obtuse. Thus, sin A ≤ sin C if C is acute or obtuse, so by the Law of Sines we have a sin A sin C a = ≤ = 1 ⇒ ≤ 1 ⇒ a ≤ c. c sin C sin C c By a similar argument, b ≤ c.