Proof of an equation by simplifying three systems of equations

Question

I am trying to prove below equality:

$A[x-i]^2+B[y-i]^2 = AB(x-y)^2$ Where $Ax+By=i$ and $A+B=1$.

But I am not able to see any obvious link to simplify this. Can anyone please help?

Answer 1

Hint: Solve $$Ax + By = i\\A + B = 1$$ for $A$ and $B$ in terms of $x, y$, first. Then substitute for them back into the first equation. Show that both sides of it can be converted to the same expression.

Then (and I cannot emphasize this enough, though early algebra students tend not to see the point), use that scratch work to write out the calculation in the proper order:

Start with left hand side, show how it reduces to that common expression, then reverse the calculation for the right hand side, to show how that common expression can be converted to the original right hand side expression. Only when you have done that reversal have you actually shown the two sides are equal.