Regiomontanus Triangle

Question

"If the base of a triangle and the opposite angle are known, and if we give or the height relative to the base or the area, then the sides can be determined". Regiomontanus proved it . How can I prove ??

Answer 1

For triangle $ABC$, suppose

• Side length $c$ is known.$\\[4pt]$
• Angle $C$ is known.$\\[4pt]$
• The area $k$ is known.

From the area formula $k=\frac{1}{2}ab\sin(C)$ we can find $ab$.

Then from the law of cosines $c^2=a^2+b^2-2ab\cos(C)$ we can find $a^2+b^2$.

Without loss of generality, assume $a\ge b$.

From the identities \begin{align*} (a+b)^2&=a^2+2ab+b^2\\[4pt] (a-b)^2&=a^2-2ab+b^2\\[4pt] \end{align*} we can find $a+b$ and $a-b$, and then we can easily solve for $a,b$.