Solve (a+b-x)^2+(b-a-y)^2=(a-b-x)^2+(a+b-y)^2

$$( a + b - x ) ^ { 2 } + ( b - a - y ) ^ { 2 } = ( a - b - x ) ^ { 2 } + ( a + b - y ) ^ { 2 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{bx}{y}\text{, }&y\neq 0\\a\in \mathrm{C}\text{, }&\left(x=0\text{ or }b=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b-x.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+\left(b-a-y\right)^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square b-a-y.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+y^{2}+2ay-2by-2ab+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine 2ab and -2ab to get 0.

$$x^{2}-2ax-2bx+a^{2}+b^{2}+y^{2}+2ay-2by+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+b^{2}+y^{2}+2ay-2by+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square a-b-x.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+\left(a+b-y\right)^{2}$$

Square a+b-y.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+y^{2}-2ay-2by+2ab+a^{2}+b^{2}$$

Combine -2ab and 2ab to get 0.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+a^{2}+b^{2}+y^{2}-2ay-2by+a^{2}+b^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+b^{2}+y^{2}-2ay-2by+b^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Add 2ax to both sides.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by+2ax=x^{2}+2bx+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Combine -2ax and 2ax to get 0.

$$x^{2}-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Subtract 2a^{2} from both sides.

$$x^{2}-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by-2a^{2}=x^{2}+2bx+2b^{2}+y^{2}-2ay-2by$$

Combine 2a^{2} and -2a^{2} to get 0.

$$x^{2}-2bx+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx+2b^{2}+y^{2}-2ay-2by$$

Add 2ay to both sides.

$$x^{2}-2bx+2b^{2}+y^{2}+2ay-2by+2ay=x^{2}+2bx+2b^{2}+y^{2}-2by$$

Combine 2ay and 2ay to get 4ay.

$$x^{2}-2bx+2b^{2}+y^{2}+4ay-2by=x^{2}+2bx+2b^{2}+y^{2}-2by$$

Subtract x^{2} from both sides.

$$-2bx+2b^{2}+y^{2}+4ay-2by=x^{2}+2bx+2b^{2}+y^{2}-2by-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-2bx+2b^{2}+y^{2}+4ay-2by=2bx+2b^{2}+y^{2}-2by$$

Add 2bx to both sides.

$$2b^{2}+y^{2}+4ay-2by=2bx+2b^{2}+y^{2}-2by+2bx$$

Combine 2bx and 2bx to get 4bx.

$$2b^{2}+y^{2}+4ay-2by=4bx+2b^{2}+y^{2}-2by$$

Subtract 2b^{2} from both sides.

$$y^{2}+4ay-2by=4bx+2b^{2}+y^{2}-2by-2b^{2}$$

Combine 2b^{2} and -2b^{2} to get 0.

$$y^{2}+4ay-2by=4bx+y^{2}-2by$$

Subtract y^{2} from both sides.

$$4ay-2by=4bx+y^{2}-2by-y^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$4ay-2by=4bx-2by$$

Add 2by to both sides.

$$4ay=4bx-2by+2by$$

Combine -2by and 2by to get 0.

$$4ay=4bx$$

Cancel out 4 on both sides.

$$ay=bx$$

The equation is in standard form.

$$ya=bx$$

Divide both sides by y.

$$\frac{ya}{y}=\frac{bx}{y}$$

Dividing by y undoes the multiplication by y.

$$a=\frac{bx}{y}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{ay}{x}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&\left(y=0\text{ or }a=0\right)\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b-x.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+\left(b-a-y\right)^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square b-a-y.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+y^{2}+2ay-2by-2ab+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine 2ab and -2ab to get 0.

$$x^{2}-2ax-2bx+a^{2}+b^{2}+y^{2}+2ay-2by+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+b^{2}+y^{2}+2ay-2by+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square a-b-x.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+\left(a+b-y\right)^{2}$$

Square a+b-y.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+y^{2}-2ay-2by+2ab+a^{2}+b^{2}$$

Combine -2ab and 2ab to get 0.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+a^{2}+b^{2}+y^{2}-2ay-2by+a^{2}+b^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+b^{2}+y^{2}-2ay-2by+b^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Subtract 2bx from both sides.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by-2bx=x^{2}-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Combine -2bx and -2bx to get -4bx.

$$x^{2}-2ax-4bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Subtract 2b^{2} from both sides.

$$x^{2}-2ax-4bx+2a^{2}+2b^{2}+y^{2}+2ay-2by-2b^{2}=x^{2}-2ax+2a^{2}+y^{2}-2ay-2by$$

Combine 2b^{2} and -2b^{2} to get 0.

$$x^{2}-2ax-4bx+2a^{2}+y^{2}+2ay-2by=x^{2}-2ax+2a^{2}+y^{2}-2ay-2by$$

Add 2by to both sides.

$$x^{2}-2ax-4bx+2a^{2}+y^{2}+2ay-2by+2by=x^{2}-2ax+2a^{2}+y^{2}-2ay$$

Combine -2by and 2by to get 0.

$$x^{2}-2ax-4bx+2a^{2}+y^{2}+2ay=x^{2}-2ax+2a^{2}+y^{2}-2ay$$

Subtract x^{2} from both sides.

$$-2ax-4bx+2a^{2}+y^{2}+2ay=x^{2}-2ax+2a^{2}+y^{2}-2ay-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-2ax-4bx+2a^{2}+y^{2}+2ay=-2ax+2a^{2}+y^{2}-2ay$$

Add 2ax to both sides.

$$-4bx+2a^{2}+y^{2}+2ay=-2ax+2a^{2}+y^{2}-2ay+2ax$$

Combine -2ax and 2ax to get 0.

$$-4bx+2a^{2}+y^{2}+2ay=2a^{2}+y^{2}-2ay$$

Subtract 2a^{2} from both sides.

$$-4bx+y^{2}+2ay=2a^{2}+y^{2}-2ay-2a^{2}$$

Combine 2a^{2} and -2a^{2} to get 0.

$$-4bx+y^{2}+2ay=y^{2}-2ay$$

Subtract y^{2} from both sides.

$$-4bx+2ay=y^{2}-2ay-y^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$-4bx+2ay=-2ay$$

Subtract 2ay from both sides.

$$-4bx=-2ay-2ay$$

Combine -2ay and -2ay to get -4ay.

$$-4bx=-4ay$$

Cancel out -4 on both sides.

$$bx=ay$$

The equation is in standard form.

$$xb=ay$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{ay}{x}$$

Dividing by x undoes the multiplication by x.

$$b=\frac{ay}{x}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{bx}{y}\text{, }&y\neq 0\\a\in \mathrm{R}\text{, }&\left(b=0\text{ or }x=0\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b-x.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+\left(b-a-y\right)^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square b-a-y.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+y^{2}+2ay-2by-2ab+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine 2ab and -2ab to get 0.

$$x^{2}-2ax-2bx+a^{2}+b^{2}+y^{2}+2ay-2by+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+b^{2}+y^{2}+2ay-2by+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square a-b-x.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+\left(a+b-y\right)^{2}$$

Square a+b-y.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+y^{2}-2ay-2by+2ab+a^{2}+b^{2}$$

Combine -2ab and 2ab to get 0.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+a^{2}+b^{2}+y^{2}-2ay-2by+a^{2}+b^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+b^{2}+y^{2}-2ay-2by+b^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Add 2ax to both sides.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by+2ax=x^{2}+2bx+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Combine -2ax and 2ax to get 0.

$$x^{2}-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Subtract 2a^{2} from both sides.

$$x^{2}-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by-2a^{2}=x^{2}+2bx+2b^{2}+y^{2}-2ay-2by$$

Combine 2a^{2} and -2a^{2} to get 0.

$$x^{2}-2bx+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx+2b^{2}+y^{2}-2ay-2by$$

Add 2ay to both sides.

$$x^{2}-2bx+2b^{2}+y^{2}+2ay-2by+2ay=x^{2}+2bx+2b^{2}+y^{2}-2by$$

Combine 2ay and 2ay to get 4ay.

$$x^{2}-2bx+2b^{2}+y^{2}+4ay-2by=x^{2}+2bx+2b^{2}+y^{2}-2by$$

Subtract x^{2} from both sides.

$$-2bx+2b^{2}+y^{2}+4ay-2by=x^{2}+2bx+2b^{2}+y^{2}-2by-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-2bx+2b^{2}+y^{2}+4ay-2by=2bx+2b^{2}+y^{2}-2by$$

Add 2bx to both sides.

$$2b^{2}+y^{2}+4ay-2by=2bx+2b^{2}+y^{2}-2by+2bx$$

Combine 2bx and 2bx to get 4bx.

$$2b^{2}+y^{2}+4ay-2by=4bx+2b^{2}+y^{2}-2by$$

Subtract 2b^{2} from both sides.

$$y^{2}+4ay-2by=4bx+2b^{2}+y^{2}-2by-2b^{2}$$

Combine 2b^{2} and -2b^{2} to get 0.

$$y^{2}+4ay-2by=4bx+y^{2}-2by$$

Subtract y^{2} from both sides.

$$4ay-2by=4bx+y^{2}-2by-y^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$4ay-2by=4bx-2by$$

Add 2by to both sides.

$$4ay=4bx-2by+2by$$

Combine -2by and 2by to get 0.

$$4ay=4bx$$

Cancel out 4 on both sides.

$$ay=bx$$

The equation is in standard form.

$$ya=bx$$

Divide both sides by y.

$$\frac{ya}{y}=\frac{bx}{y}$$

Dividing by y undoes the multiplication by y.

$$a=\frac{bx}{y}$$

Solve for b

$$\left\{\begin{matrix}b=\frac{ay}{x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&\left(a=0\text{ or }y=0\right)\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Square a+b-x.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+\left(b-a-y\right)^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square b-a-y.

$$x^{2}-2ax-2bx+2ab+a^{2}+b^{2}+y^{2}+2ay-2by-2ab+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine 2ab and -2ab to get 0.

$$x^{2}-2ax-2bx+a^{2}+b^{2}+y^{2}+2ay-2by+a^{2}+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+b^{2}+y^{2}+2ay-2by+b^{2}=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=\left(a-b-x\right)^{2}+\left(a+b-y\right)^{2}$$

Square a-b-x.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+\left(a+b-y\right)^{2}$$

Square a+b-y.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax-2ab+a^{2}+b^{2}+y^{2}-2ay-2by+2ab+a^{2}+b^{2}$$

Combine -2ab and 2ab to get 0.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+a^{2}+b^{2}+y^{2}-2ay-2by+a^{2}+b^{2}$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+b^{2}+y^{2}-2ay-2by+b^{2}$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}+2bx-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Subtract 2bx from both sides.

$$x^{2}-2ax-2bx+2a^{2}+2b^{2}+y^{2}+2ay-2by-2bx=x^{2}-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Combine -2bx and -2bx to get -4bx.

$$x^{2}-2ax-4bx+2a^{2}+2b^{2}+y^{2}+2ay-2by=x^{2}-2ax+2a^{2}+2b^{2}+y^{2}-2ay-2by$$

Subtract 2b^{2} from both sides.

$$x^{2}-2ax-4bx+2a^{2}+2b^{2}+y^{2}+2ay-2by-2b^{2}=x^{2}-2ax+2a^{2}+y^{2}-2ay-2by$$

Combine 2b^{2} and -2b^{2} to get 0.

$$x^{2}-2ax-4bx+2a^{2}+y^{2}+2ay-2by=x^{2}-2ax+2a^{2}+y^{2}-2ay-2by$$

Add 2by to both sides.

$$x^{2}-2ax-4bx+2a^{2}+y^{2}+2ay-2by+2by=x^{2}-2ax+2a^{2}+y^{2}-2ay$$

Combine -2by and 2by to get 0.

$$x^{2}-2ax-4bx+2a^{2}+y^{2}+2ay=x^{2}-2ax+2a^{2}+y^{2}-2ay$$

Subtract x^{2} from both sides.

$$-2ax-4bx+2a^{2}+y^{2}+2ay=x^{2}-2ax+2a^{2}+y^{2}-2ay-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-2ax-4bx+2a^{2}+y^{2}+2ay=-2ax+2a^{2}+y^{2}-2ay$$

Add 2ax to both sides.

$$-4bx+2a^{2}+y^{2}+2ay=-2ax+2a^{2}+y^{2}-2ay+2ax$$

Combine -2ax and 2ax to get 0.

$$-4bx+2a^{2}+y^{2}+2ay=2a^{2}+y^{2}-2ay$$

Subtract 2a^{2} from both sides.

$$-4bx+y^{2}+2ay=2a^{2}+y^{2}-2ay-2a^{2}$$

Combine 2a^{2} and -2a^{2} to get 0.

$$-4bx+y^{2}+2ay=y^{2}-2ay$$

Subtract y^{2} from both sides.

$$-4bx+2ay=y^{2}-2ay-y^{2}$$

Combine y^{2} and -y^{2} to get 0.

$$-4bx+2ay=-2ay$$

Subtract 2ay from both sides.

$$-4bx=-2ay-2ay$$

Combine -2ay and -2ay to get -4ay.

$$-4bx=-4ay$$

Cancel out -4 on both sides.

$$bx=ay$$

The equation is in standard form.

$$xb=ay$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{ay}{x}$$

Dividing by x undoes the multiplication by x.

$$b=\frac{ay}{x}$$