Question

If $a,b,c,d$ are proportional, prove that $a+d=b+c+\frac{(a-b)(a-c)}{a}$.

Answer 1

$\frac{a}{b}=\frac{c}{d}$ So, $d=\frac{bc}{a}$

Now,

$b+c+\frac{(a-b)(a-c)}{a}$

$=\frac{ab+ac+a^2-ac-ab+bc}{a}$

$=\frac{a^2+bc}{a}=a+\frac{bc}{a}=a+d$