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amoodarya
 one year ago
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
$$s=\sqrt{p(pa)(pb)(pc)} ,p=\frac{a+b+c}{2}\\s=\frac{h_a*a}{2}\\s=\frac{1}{2}bc\sin(A)\\s=2R^2\sin A \sin B \sin C$$
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
amoodarya
 one year ago
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 $$s=\sqrt{p(pa)(pb)(pc)} ,p=\frac{a+b+c}{2}\\s=\frac{h_a*a}{2}\\s=\frac{1}{2}bc\sin(A)\\s=2R^2\sin A \sin B \sin C$$ 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

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amoodarya
 one year ago
Best ResponseYou've already chosen the best response.3I 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 $$s=\sqrt{p(pa)(pb)(pc)} ,p=\frac{a+b+c}{2}\\s=\frac{h_a*a}{2}\\s=\frac{1}{2}bc\sin(A)\\s=2R^2\sin A \sin B \sin C$$ 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 2d 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: $$ s=2R^2sin(\frac{p}{2})sin(\frac{q}{2})sin(\frac{pq}{2})$$ \[s=\frac{abc}{4R}\\s=rp\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0the 2nd and 3rd are identical, one just computes the altitude \(h_a\) using trig

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0also 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
 one year ago
Best ResponseYou've already chosen the best response.0the 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
 one year ago
Best ResponseYou've already chosen the best response.3I see this page , you give I see it for the first time ! I collect them from my note (s) in many years

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0hmm, well whatever source you collected these notes contains wordforword copied sections? http://puu.sh/jalaS/288242df75.png http://puu.sh/jale2/7622d692ec.png

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.3wow ! most of them was in "math passages in english " when I was b.s student it was many years ago ...
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