S L Loney Solutions for Chapter: Properties of Triangle, Exercise 2: Examples XXXVI

In triangle $ABC$, prove that $4R\sin A\sin B\sin C=a\cos A+b\cos B+c\cos C$, where $a, b \& c$ are the sides of the triangle and $R$ is the circumradius.

Prove that in triangle $\mathrm{ABC}$, $S=4Rr\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$, where $r$ is radius of incircle, $R$ is radius of circumcircle and $S$ is area of triangle.

Prove that in triangle $ABC$, $\frac{1}{r^2}+\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}=\frac{a^2+b^2+c^2}{S^2}$, where $a, b \& c$ are the lengths sides of the triangle$△ABC$  and $r$ is radius of incircle, $r_1, r_2 \& r_3$ are the radius of excircles and $S$ is area of triangle.

Prove that in triangle $\mathrm{ABC}$, $r_1+r_2+r_3-r=4R$, where $R$ is radius of circumcircle and $\mathrm{r}$ is radius of incircle, $r_1, r_2 \& r_3$ are the radii of the excircles.

Prove that in triangle $ABC$, $\left(r_1-r\right)\left(r_2-r\right)\left(r_3-r\right)=4R{r}^2$, where $R$ is radius of circumcircle and $r$ is radius of incircle, $r_1, r_2 \& r_3$ are the radii of the excircles.

Prove that in triangle $ABC$, $\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab}=\frac{1}{2Rr}$, where $R$ is circumradius of the circumcircle and $r$ is the inradius of incircle, $a, b \& c$ are the sides of the triangle $△ABC$.

Prove that in triangle $ABC$, $\frac{r_1}{bc}+\frac{r_2}{ca}+\frac{r_3}{ab}=\frac{1}{r}-\frac{1}{2R}$, where $R$ is radius of circumcircle and $r_1, r_2 \& r_3$ are the radius of excircles and $r$ is radius of incircle, $a, b \& c$ are the sides of the triangle.

Prove that in triangle $ABC$, $r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2$, where $R$ is radius of circumcircle, $r_1, r_2 \& r_3$ are the radius of excircles, $r$ is radius of incircle and $a, b \& c$ are the sides of the triangle.