Derived From De Gua's Theorem - Works like Magic !!!
5 Coexisting Relationships BUT ONLY in TriRectangular Tetrahedrons
![{\displaystyle \ \ \ \ K={\Biggl [}{\frac {a^{2}+b^{2}+c^{2}}{2}}{\Biggr ]}={\Bigl [}x^{2}+y^{2}+z^{2}{\Bigr ]}={\Bigl [}x^{2}+c^{2}{\Bigr ]}={\Bigl [}y^{2}+b^{2}{\Bigr ]}={\Bigl [}z^{2}+a^{2}{\Bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6063cc38d90bd1637abeade9b4962d8a899e2fe4)
Perfect Mirrored Imaged Equations
![{\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Given{\Bigl [}a,b,c{\Bigr ]}\ \ \ \ \ \ \ \ \ \ \ \ \ Given{\Bigl [}x,y,z{\Bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2aba5351c9158ce4279a97320536f1f5de2eb1)
![{\displaystyle \ \ \ \ \ K={\Bigl [}Constant{\Bigr ]}\ \ \ \ \ \ \ \ \ K={\Biggl [}{\frac {a^{2}+b^{2}+c^{2}}{2}}{\Biggr ]}\ \ \ \ \ \ \ \ \ K={\Bigl [}x^{2}+y^{2}+z^{2}{\Bigr ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bae0b58fa7b3d359722b9e7d55672907422218f7)
![{\displaystyle \ \ \ \ Use\ {\Bigl [}x^{2}+c^{2}{\Bigr ]}\ \ \ \ \ \ \ \ x={\sqrt {_{\ }K-c^{2}\ \ }}\ \ \ \ \ \ \ \ \ \ \ c={\sqrt {_{\ }K-x^{2}\ \ }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d39edb16ab75510bd9e1b0f538ef9e1f6c68c2e)
![{\displaystyle \ \ \ \ Use\ {\Bigl [}y^{2}+b^{2}{\Bigr ]}\ \ \ \ \ \ \ \ \ y={\sqrt {_{\ }K-b^{2}\ \ }}\ \ \ \ \ \ \ \ \ \ \ b={\sqrt {_{\ }K-y^{2}\ \ }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e215a368b3d320ad2fd75bad927228b6b0099a9d)
![{\displaystyle \ \ \ \ Use\ {\Bigl [}z^{2}+a^{2}{\Bigr ]}\ \ \ \ \ \ \ \ \ z={\sqrt {_{\ }K-a^{2}\ \ }}\ \ \ \ \ \ \ \ \ \ \ a={\sqrt {_{\ }K-z^{2}\ \ }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05e35b3debe2c9c3f1b3174385831cbefaec3ed3)
Volume of Tetrahedron & Enclosing Box
Height of Tetrahedron
![{\displaystyle \ \ \ \ \ h_{tet}\ ={\frac {xyz}{\ \ {\sqrt {\ x^{2}y^{2}+z^{2}{\Bigl [}x^{2}+y^{2}{\Bigr ]}\ }}\ \ }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/926d1299604582a21e6b1e3fee7f3405740d902a)
2 Formulas For The Area of The Base
![{\displaystyle \ \ \ \ A_{abc}={\frac {xyz}{\ 2h_{tet}}}\ \ \ \ \ \ \ A_{abc}={\frac {\ \ {\sqrt {x^{2}y^{2}+z^{2}{\Bigl [}x^{2}+y^{2}{\Bigr ]}\ }}\ \ \ }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c856eefdd2ac5ecbf72c20b14dea51542ea52da8)
Individual Areas of All Four Surfaces
Total Surface Area of a Tetrahedron
![{\displaystyle \ \ \ \ A_{tet}={\frac {\ xy+z{\Bigl [}x+y{\Bigr ]}+{\sqrt {x^{2}y^{2}+z^{2}{\Bigl [}x^{2}+y^{2}{\Bigr ]}\ }}\ \ }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d43713003b787f576d0b025b7be2ab19528267)
Example Parameters
The Other Way - Harder to Use But Still Works
![{\displaystyle Given{\Bigl [}x,y,z{\Bigr ]}\ \ \ \ \ \ \ \ a={\sqrt {\ x^{2}+y^{2}\ }}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b={\sqrt {\ x^{2}+z^{2}\ }}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c={\sqrt {\ y^{2}+z^{2}\ }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/365a5a15b82253ebac911767b1a81994787e6d0a)
![{\displaystyle Given\ {\Bigl [}a,b,c{\Bigr ]}\ \ \ \ \ \ \ \ x={\sqrt {{\frac {\ a^{2}+b^{2}-c^{2}\ }{2}}\ }}\ \ \ \ \ \ \ \ y={\sqrt {{\frac {\ a^{2}+c^{2}-b^{2}\ }{2}}\ }}\ \ \ \ \ \ \ \ z={\sqrt {{\frac {\ b^{2}+c^{2}-a^{2}\ }{2}}\ }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3def812c82aaef689ec864b05230bc5772c5e263)
