Solve AM^2+BC^2=AC^2+BM^2

$$A { M }^{ 2 } +B { C }^{ 2 } = A { C }^{ 2 } +B { M }^{ 2 }$$

Solve for A

$$\left\{\begin{matrix}\\A=B\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&|M|=|C|\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract AC^{2} from both sides.

$$AM^{2}+BC^{2}-AC^{2}=BM^{2}$$

Subtract BC^{2} from both sides.

$$AM^{2}-AC^{2}=BM^{2}-BC^{2}$$

Combine all terms containing A.

$$\left(M^{2}-C^{2}\right)A=BM^{2}-BC^{2}$$

Divide both sides by M^{2}-C^{2}.

$$\frac{\left(M^{2}-C^{2}\right)A}{M^{2}-C^{2}}=\frac{B\left(M-C\right)\left(C+M\right)}{M^{2}-C^{2}}$$

Dividing by M^{2}-C^{2} undoes the multiplication by M^{2}-C^{2}.

$$A=\frac{B\left(M-C\right)\left(C+M\right)}{M^{2}-C^{2}}$$

Divide B\left(M-C\right)\left(M+C\right) by M^{2}-C^{2}.

$$A=B$$

Solve for B

$$\left\{\begin{matrix}\\B=A\text{, }&\text{unconditionally}\\B\in \mathrm{R}\text{, }&|M|=|C|\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract BM^{2} from both sides.

$$AM^{2}+BC^{2}-BM^{2}=AC^{2}$$

Subtract AM^{2} from both sides.

$$BC^{2}-BM^{2}=AC^{2}-AM^{2}$$

Combine all terms containing B.

$$\left(C^{2}-M^{2}\right)B=AC^{2}-AM^{2}$$

Divide both sides by C^{2}-M^{2}.

$$\frac{\left(C^{2}-M^{2}\right)B}{C^{2}-M^{2}}=\frac{A\left(C-M\right)\left(C+M\right)}{C^{2}-M^{2}}$$

Dividing by C^{2}-M^{2} undoes the multiplication by C^{2}-M^{2}.

$$B=\frac{A\left(C-M\right)\left(C+M\right)}{C^{2}-M^{2}}$$

Divide A\left(C-M\right)\left(C+M\right) by C^{2}-M^{2}.

$$B=A$$