Help me complete the solution to a geometry problem

Question

This is the solution given in my book but it is incomplete. What will be the next steps to complete the solution . Edit- After simplifying the problem I now have to prove this inequality . ((x+y+z)^2)>=6xyz How do i prove this inequality ?

Answer 1

Each of the terms in the square brackets are the type $a+1/a$. For example, in the first one $$a=\frac xy\\\frac1a=\frac yx$$ You can use AM-GM inequality$$\frac{a+\frac1a}2\ge\sqrt{a\cdot \frac1a}\\a+\frac1a\ge 2$$ The equality happens when $a=1/a$ or $a=1$. Putting it all together $$\left[\frac xy+\frac yx\right]+\left[\frac yz+\frac zy\right]+\left[\frac zx+\frac xz\right]\ge 2+2+2=6$$ You get $6$ when $x=y=z$.