<s>[INST] Proof of $ x^2 + y = y^2 + x$ when $ x+ y =1$ and $x$ is larger than $y$ I know that whatever numbers you choose for x and y and their sum equals to 1 will satisfy the equation $x^2 + y = y^2 + x$
Algebraic proof: 
Given:   $x + y = 1$
$$LS = x^2+ y 
= (1-y)^2 + y 
= 1 - 2y+y^2 + y
= y^2 - y + 1$$
$$RS = y^2 + x
= y^2 + (1-y)
= y^2 - y + 1$$
Therefore,$$ LS = RS $$
How can this be proved geometrically?  (Ex. in a diagram of rectangular areas)
I tried to add a square piece with side lengths y with a rectangle with side lengths x and x+y but I can't seem to prove it geometrically. 
Can someone help? 
 [/INST] Because $$0=x^2+y-(y^2+x)=(x-y)(x+y)-(x-y)=(x-y)(x+y-1).$$
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