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Question Number 40469 by MrW3 last updated on 22/Jul/18

Commented by MrW3 last updated on 22/Jul/18

$$Findthevolumeofpyramid.$$

Commented by MJS last updated on 22/Jul/18

$${\mathrm{Euler}}^{\prime }\mathrm{s}\mathrm{formula}:$$$$\frac{1}{12}\sqrt{a^2{p}^2{f}_a+b^2{q}^2{f}_b+c^2{r}^2{f}_c-\delta }$$$$f_a=-(a^2+p^2)+(b^2+q^2)+(c^2+r^2)$$$$f_b=(a^2+p^2)-(b^2+q^2)+(c^2+r^2)$$$$f_c=(a^2+p^2)+(b^2+q^2)-(c^2+r^2)$$$$\delta =a^2{b}^2{c}^2+a^2{q}^2{r}^2+b^2{p}^2{r}^2+c^2{p}^2{q}^2$$

Commented by MrW3 last updated on 22/Jul/18

$$thankyousir!$$

Commented by MJS last updated on 22/Jul/18

$${\mathrm{there}}^{\prime }\mathrm{s}\mathrm{also}\mathrm{a}\mathrm{matrix}-\mathrm{formula}\mathrm{for}\mathrm{a}\mathrm{simplex}$$$$\mathrm{in}{\mathbb{R}}^n(\mathrm{in}{\mathbb{R}}^1\mathrm{the}\mathrm{simplex}\mathrm{is}\mathrm{a}\mathrm{line}\mathrm{segment},\mathrm{in}$$$${\mathbb{R}}^2\mathrm{a}\mathrm{triangle}\mathrm{and}\mathrm{in}{\mathbb{R}}^3\mathrm{a}\mathrm{tetrahedron},\mathrm{the}$$$$\mathrm{Euler}-\mathrm{formula}\mathrm{for}{\mathbb{R}}^2\mathrm{is}\mathrm{the}\mathrm{same}\mathrm{as}{\mathrm{Heron}}^{\prime }\mathrm{s}$$$$A_{△abc}=\frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)})$$$$\mathrm{but}\mathrm{I}{\mathrm{couldn}}^{\prime }\mathrm{t}\mathrm{find}\mathrm{this}\mathrm{matrix}...$$

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