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

Question

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

amoodarya · Answer

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 · Answer

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amoodarya · Answer

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anonymous · Answer

the 2nd and 3rd are identical, one just computes the altitude $h_a$ using trig

anonymous · Answer

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 · Answer

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 · Answer

I see this page , you give 
I see it for the first time !
I collect them from my note (s) in many years

anonymous · Answer

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 · Answer

wow !
most of them was in "math passages in english "
when I was b.s student 
it was many years ago ...