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Let and . Prove that the equation
has no solution.
Without loss of generality , assume that . If held , then it would be thus . It follows from Bernoulli’s inequality that,
which is an obscurity. The result follows.
Evaluate the double sum
We sum diagonally , hence:
(1): For it holds .
Conjecture: Does the following equality
Let be a triangle such that , and . Find the area of the triangle.
Since it follows from the law of sines that
Hence . Thus,
To completely justify the title of the post we give another solution based on the following proposition:
Proposition: Let be a given triangle such that . Prove that
Proof: We are working on the following figure.
Let be the bisector of . Then:
and the result follows.
Now, the area of the initial triangle is given by Heron’s formula.
Let . Prove that
Lemma: If or then it holds