viernes, 7 de mayo de 2021

Using the half-angle formulas to derive Mahavira's identities

In a cyclic quadrilateral $ABCD$, let $a$, $b$, $c$, $d$ denote the lengths of sides $AB$, $BC$, $CD$, $DA$, and $m$, $n$ the lengths of the diagonals $BD$ and $BC$. Then Mahavira's result is expressed as

Proof. By the Law of Cosines, 


Substituting from the half-angle formula (see formula $(5)$ in this page) we get


Similarly we can get $(2)$.

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