Question 50
The center of gravity of triangle $ABC$ is $G$. Let $g_{a}, g_{b}, g_{c}$ denote the distances from $G$ to the sides $a, b, c$, respectively. Let $r$ be the radius of the inscribed circle.
a) Prove that
$$
g_{a} \geq \frac{2r}{3}, \quad g_{b} \geq \frac{2r}{3}, \quad g_{c} \geq \frac{2r}{3}
$$
b) Prove that
$$
\frac{g_{a}+g_{b}+g_{c}}{r} \geq 3
$$
a) Prove that
$$
g_{a} \geq \frac{2r}{3}, \quad g_{b} \geq \frac{2r}{3}, \quad g_{c} \geq \frac{2r}{3}
$$
b) Prove that
$$
\frac{g_{a}+g_{b}+g_{c}}{r} \geq 3
$$