Solve {l}{B^{2}-BH^{2}=AC^{2}-CH^{2}(=AH^{2})}{20^2-x^2=13^2-(21-x)^2,x=16}

$$\left. \begin{array} { l } { B ^ { 2 } - BH ^ { 2 } = AC ^ { 2 } - CH ^ { 2 } ( = AH ^ { 2 } ) } \\ { 20 ^ { 2 } - x ^ { 2 } = 13 ^ { 2 } - ( 21 - x ) ^ { 2 } , x = 16 } \end{array} \right.$$

Solve for B, H, A, C

$$\left\{\begin{matrix}\\B=\frac{\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H\in \mathrm{C}\setminus \sqrt{A}\sqrt{C},-\sqrt{A}\sqrt{C}\text{, }A\in \mathrm{C}\text{, }C\neq 0\text{; }B=\frac{-\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H\in \mathrm{C}\setminus \sqrt{A}\sqrt{C},-\sqrt{A}\sqrt{C}\text{, }A\in \mathrm{C}\text{, }C\neq 0\text{; }B=0\text{, }H=0\text{, }A=0\text{, }C\in \mathrm{C}\text{; }B=0\text{, }H=0\text{, }A\in \mathrm{C}\text{, }C=0\text{, }&\text{unconditionally}\\B=\frac{\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H\in \mathrm{C}\text{, }A\in \mathrm{C}\text{, }C\in \mathrm{C}\text{, }&|\frac{arg(H^{4})}{2}-arg(-H^{2})|\geq \pi \text{ and }H\neq 0\\B=\frac{\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H=\sqrt{A}\sqrt{C}\text{, }A\in \mathrm{C}\text{, }C\neq 0\text{; }B=\frac{\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H=-\sqrt{A}\sqrt{C}\text{, }A\in \mathrm{C}\text{, }C\neq 0\text{, }&H\neq 0\text{ and }arg(H^{2})\geq \pi \text{ and }A\neq 0\text{ and }arg(AC)\geq \pi \\B=H^{2}\text{, }H\in \mathrm{C}\text{, }A\in \mathrm{C}\text{, }C=0\text{; }B=\frac{-\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H\in \mathrm{C}\text{, }A\in \mathrm{C}\text{, }C\in \mathrm{C}\text{, }&H\neq 0\text{ and }arg(H^{2})\geq \pi \\B=\frac{-\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H=\sqrt{A}\sqrt{C}\text{, }A\in \mathrm{C}\text{, }C\neq 0\text{; }B=\frac{-\sqrt{4AC^{2}-4CH^{2}+H^{4}}+H^{2}}{2}\text{, }H=-\sqrt{A}\sqrt{C}\text{, }A\in \mathrm{C}\text{, }C\neq 0\text{, }&H\neq 0\text{ and }arg(H^{2})<\pi \text{ and }A\neq 0\text{ and }arg(AC)<\pi \\B=0\text{, }H\in \mathrm{C}\text{, }A\in \mathrm{C}\text{, }C=0\text{, }&H\neq 0\text{ and }arg(H^{2})<\pi \end{matrix}\right.$$