Show $\cos\frac\gamma2=\sqrt{\frac{p(p-c)}{ab}}$ 
Show that the equality $$\cos\dfrac{\gamma}{2}=\sqrt{\dfrac{p(p-c)}{ab}}$$ holds for a $\triangle ABC$ with sides $AB=c,BC=a, AC=b$, semi-perimeter $p$ and $\measuredangle ACB=\gamma$.


I have just proved that $$l_c=\dfrac{2ab\cos\dfrac{\gamma}{2}}{a+b}$$
Is this a well known formula for angle bisectors?
Can we use it somehow?
Thank you in advance!
 A: Form half-angle trig-identity
$$\cos\gamma=2\cos^2\frac{\gamma}{2}-1\iff \cos \frac{\gamma}{2}=\sqrt{\frac{1+\cos\gamma}{2}}$$
Substituting the value from cosine formula: $\cos \gamma=\frac{a^2+b^2-c^2}{2ab}$
$$\cos \frac{\gamma}{2}=\sqrt{\frac{1+\frac{a^2+b^2-c^2}{2ab}}{2}}$$
$$=\sqrt{\frac{(a+b)^2-c^2}{4ab}}$$
$$=\sqrt{\frac{(a+b+c)(a+b-c)}{4ab}}$$
$$=\sqrt{\frac{\left(\frac{a+b+c}{2}\right)\left(\frac{a+b+c}{2}-c\right)}{ab}}$$
$$=\sqrt{\frac{p(p-c)}{ab}}$$
A: We have standard formula for $R = {abc\over 4S}$ and $S = rp$, where $S$ is area and $p$ perimeter of triangle.


*

*$\cot {\gamma \over 2} = {p-c\over r}$

*$\sin \gamma  = {c\over 2R}$ (the law of sine)

If we multiply these two we get $$\cos ^2 {\gamma\over 2} = {S(p-c)\over rab} = {p(p-c)\over ab}$$
(Remember that $\sin \gamma = 2\sin{\gamma\over 2}\cos {\gamma \over 2}$)
A: Your formula for length of angle bisectors is correct. To derive for sine/cosine of half-angles, a straightforward way is using the formulas : $$1+\cos 2\theta = 2\cos^2 \theta \, , \quad 1-\cos 2\theta = 2\sin^2 \theta$$
By cosine-rule in $\triangle ABC$,
$$\cos \gamma = \frac{a^2+b^2-c^2}{2ab}$$
$$\Rightarrow \cos \frac{\gamma}{2}=\sqrt{\frac{1+\cos \gamma}{2}}$$
which simplifies to
$$ \cos \frac{\gamma}{2}=\sqrt{\frac{p(p-c)}{ab}}$$
Similarly you can also find $\sin (\gamma/2)$, $\tan (\gamma/2)$ etc.
A: As for the last part (how we can use the bisector).. The property that the ratio of sides is the ratio in which opposite side $c$ is intersected
$$ e=\dfrac{AC}{ CB}=\dfrac{  AL}{  LB},$$
a constant that defines the Apollonius circle of constant $e$. Bisection can be done either for the internal or for external angle.
