Question 191
Let $ABC$ be a triangle with sides $a, b, c$. Consider a triangle $A_1 B_1 C_1$ with sides equal to $a+\frac{b}{2}$, $b+\frac{c}{2}$, $c+\frac{a}{2}$. Show that
$$
\left[A_1 B_1 C_1\right] \geq \frac{9}{4}[ABC]
$$
where $[XYZ]$ denotes the area of the triangle $XYZ$.
$$
\left[A_1 B_1 C_1\right] \geq \frac{9}{4}[ABC]
$$
where $[XYZ]$ denotes the area of the triangle $XYZ$.