Mathematics - Properties of Triangle Question with Solution | TestHub

Let the triangle have area $\Delta$ and base $a$, and let the bases of the three small triangles along BC be $x_1,x_2,x_3$ with $x_1+x_2+x_3=a$. The small triangles are similar to ABC, so $\sqrt{\frac{{\Delta }_1}{\Delta }}=\frac{x_1}{a},\sqrt{\frac{{\Delta }_2}{\Delta }}=\frac{x_2}{a},\sqrt{\frac{{\Delta }_3}{\Delta }}=\frac{x_3}{a}$. Adding, $\sqrt{\Delta }=\sqrt{{\Delta }_1}+\sqrt{{\Delta }_2}+\sqrt{{\Delta }_3}$, so $\Delta ={\Delta }_1+{\Delta }_2+{\Delta }_3+2\left(\sqrt{{\Delta }_1{\Delta }_2}+\sqrt{{\Delta }_2{\Delta }_3}+\sqrt{{\Delta }_3{\Delta }_1}\right)$