Prove the identity of a triangle in the given problem

AI Thread Summary
The discussion focuses on proving the identity of a triangle using cosine relationships and the semi-perimeter. The approach begins with the equation for cosine squared of half angles and derives a relationship involving the sides of the triangle and its semi-perimeter. It further simplifies the expression to show that the sum of the cosine squared terms equals a specific fraction involving the semi-perimeter and the sides of the triangle. Both proofs presented in the thread utilize similar principles and lead to the same conclusion. The discussion invites insights or alternative methods for proving the identity.
chwala
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Homework Statement
Prove that in a triangle ABC

##\dfrac{1}{a} \cos^2 \dfrac{1}{2} A+ \dfrac{1}{b} \cos^2 \dfrac{1}{2} B +\dfrac{1}{c} \cos^2 \dfrac{1}{2} C = \dfrac{(a+b+c)^2}{abc}##
Relevant Equations
Trig. identities
My approach,
##\cos^2 \frac{1}{2} A = \dfrac{s(s-a)}{bc}## ...
then it follows that,

##\dfrac{1}{a} \cos^2 \dfrac{1}{2} A+ \dfrac{1}{b} \cos^2 \dfrac{1}{2} B +\dfrac{1}{c} \cos^2 \dfrac{1}{2} C =\dfrac{s(s-a)+s(s-b)+s(s-c)}{abc}##

##=\dfrac {3s^2-s(a+b+c)}{abc}##

##=\dfrac{3s^2-2s^2}{abc}##

##=\dfrac{s^2}{abc}=\dfrac{(a+b+c)^2}{4abc}##

any insight or better approach welcome.
 
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\frac{1}{a}\cos ^2\frac{A}{2}=\frac{\cos A}{2a}+\frac{1}{2a}=\frac{b^2+c^2-a^2}{4abc}+\frac{1}{2a}
using cosine theorem
a^2=b^2+c^2-2bc \cos A
Doing the same for b and c, the sum is
\frac{a^2+b^2+c^2}{4abc}+\frac{1}{2a}+\frac{1}{2b}+\frac{1}{2c}=\frac{(a+b+c)^2}{4abc}
 
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Likes neilparker62, SammyS, scottdave and 2 others
## \begin{align}
\cos^2\frac A2&=\frac {s(s-a)}{bc}\nonumber\\
&=\frac{(a+b+c)(b+c-a)}{4bc}\nonumber\\
&=\frac{ab+b^2+bc+ac+bc+c^2-a^2-ab-ac}{4bc}\nonumber\\
&=\frac{b^2+c^2-a^2}{4bc}+\frac 12\nonumber
\end{align} ##

The proof in the post #1 and the proof in the post #2 are almost the same. They use the same principle.
 
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