amoodarya
  • amoodarya
I know some formulas to find a triangle's area, like the ones below. 1. Is there any reference containing most triangle area formulas? 1. If you know more, please add them as an answer $$s=\sqrt{p(p-a)(p-b)(p-c)} ,p=\frac{a+b+c}{2}\\s=\frac{h_a*a}{2}\\s=\frac{1}{2}bc\sin(A)\\s=2R^2\sin A \sin B \sin C$$ Another symmetrical form is given by :$$(4s)^2=\begin{bmatrix} a^2 & b^2 & c^2 \end{bmatrix}\begin{bmatrix} -1 & 1 & 1\\ 1 & -1 & 1\\ 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} a^2\\ b^2\\ c^2 \end{bmatrix}$$ [![triangle with three mutuaally tangent circles centr
Mathematics
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amoodarya
  • amoodarya
I know some formulas to find a triangle's area, like the ones below. 1. Is there any reference containing most triangle area formulas? 1. If you know more, please add them as an answer $$s=\sqrt{p(p-a)(p-b)(p-c)} ,p=\frac{a+b+c}{2}\\s=\frac{h_a*a}{2}\\s=\frac{1}{2}bc\sin(A)\\s=2R^2\sin A \sin B \sin C$$ Another symmetrical form is given by :$$(4s)^2=\begin{bmatrix} a^2 & b^2 & c^2 \end{bmatrix}\begin{bmatrix} -1 & 1 & 1\\ 1 & -1 & 1\\ 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} a^2\\ b^2\\ c^2 \end{bmatrix}$$ [![triangle with three mutuaally tangent circles centr
Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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amoodarya
  • amoodarya
I know some formulas to find a triangle's area, like the ones below. 1. Is there any reference containing most triangle area formulas? 1. If you know more, please add them as an answer $$s=\sqrt{p(p-a)(p-b)(p-c)} ,p=\frac{a+b+c}{2}\\s=\frac{h_a*a}{2}\\s=\frac{1}{2}bc\sin(A)\\s=2R^2\sin A \sin B \sin C$$ Another symmetrical form is given by :$$(4s)^2=\begin{bmatrix} a^2 & b^2 & c^2 \end{bmatrix}\begin{bmatrix} -1 & 1 & 1\\ 1 & -1 & 1\\ 1 & 1 & -1 \end{bmatrix} \begin{bmatrix} a^2\\ b^2\\ c^2 \end{bmatrix}$$ [![triangle with three mutuaally tangent circles centred on the vertices][1]][1] Expressing the side lengths $a$, $b$ & $c$ in terms of the radii $a'$, $b'$ & $c'$ of the mutually tangent circles centered on the triangle's vertices (which define the Soddy circles) $$a=b'+c'\\b=a'+c'\\c=a'+b'$$gives the paticularly pretty form $$s=\sqrt{a'b'c'(a'+b'+c')}$$ If the triangle is embedded in three dimensional space with the coordinates of the vertices given by $(x_i,y_i,z_i)$ then $$s=\frac{1}{2}\sqrt{\begin{vmatrix} y_1 &z_1 &1 \\ y_2&z_2 &1 \\ y_3 &z_3 &1 \end{vmatrix}^2+\begin{vmatrix} z_1 &x_1 &1 \\ z_2&x_2 &1 \\ z_3 &x_3 &1 \end{vmatrix}^2+\begin{vmatrix} x_1 &y_1 &1 \\ x_2&y_2 &1 \\ x_3 &y_3 &1 \end{vmatrix}^2}$$ When we have 2-d coordinate $$ s=\frac{1}{2}\begin{vmatrix} x_a &y_a &1 \\ x_b &y_b &1 \\ x_c &y_c & 1 \end{vmatrix}$$ [![enter image description here][2]][3] In the above figure, let the circumcircle passing through a triangle's vertices have radius $R$, and denote the central angles from the first point to the second $q$, and to the third point by $p$ then the area of the triangle is given by: $$ s=2R^2|sin(\frac{p}{2})sin(\frac{q}{2})sin(\frac{p-q}{2})|$$ \[s=\frac{abc}{4R}\\s=rp\]
amoodarya
  • amoodarya
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anonymous
  • anonymous
the 2nd and 3rd are identical, one just computes the altitude \(h_a\) using trig
anonymous
  • anonymous
also why would you bother typing out all of this if you're just copying things from a mathworld article? http://mathworld.wolfram.com/TriangleArea.html
anonymous
  • anonymous
the symmetrical form with a matrix is just because the expression under the radical is a quadratic form in \(a^2,b^2,c^2\), and in general a quadratic form \(Q\) is expressible as a matrix product \(x^T A x\) where \(A\) is a symmetric matrix
amoodarya
  • amoodarya
I see this page , you give I see it for the first time ! I collect them from my note (s) in many years
anonymous
  • anonymous
hmm, well whatever source you collected these notes contains word-for-word copied sections? http://puu.sh/jalaS/288242df75.png http://puu.sh/jale2/7622d692ec.png
amoodarya
  • amoodarya
wow ! most of them was in "math passages in english " when I was b.s student it was many years ago ...

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