lunes, 27 de junio de 2022

A generalization of the Pythagorean trigonometric identity

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"Discovery is seeing what everybody else has seen, and thinking what nobody else has thought." -- Albert Szent-Györgyi (1893 - 198...
sábado, 25 de junio de 2022

Another proof of Euler inequality via the half-angle formulas

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The Euler's inequality  is an immediate consequence of  Euler's identity in a triangle , $$OI^2=R^2−2Rr.$$ An additional proof of Eu...
domingo, 19 de junio de 2022

$\sum_{cyc}\tan\frac\alpha2\tan\frac\beta2\geq4$ for a cyclic quadrilateral

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 Let $ABCD$ be a cyclic quadrilateral with sides $a$, $b$, $c$ and $d$. Denote $s$ the semiperimeter and let $\angle{DAB}=\alpha$, $\angle{A...
lunes, 23 de mayo de 2022

The theoretical importance of half-angle formulas

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"The scientific principles that man employs to obtain the foreknowledge of an eclipse, or of any thing else relating to the motion of t...
domingo, 22 de mayo de 2022

La importancia teórica de las fórmulas de medio ángulo

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  A diferencia de las leyes  de senos ,  de cosenos y  de las tangentes , que son muy bien conocidas, las fórmulas de medio ángulo parecen ...
miércoles, 27 de abril de 2022

The compound angle formulas from the half angle formulas

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My goal is not necessarily to give the simplest derivation. I've been trying to rescue the  half angle formulas from oblivion and give ...
domingo, 27 de marzo de 2022

A generalization of the law of tangent for a cyclic quadrilateral

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The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing s...