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Math formulas: Triangles in two dimensions

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Area of the triangle

The area of the triangle formed by the three lines:

A1x+B1y+C1=0A2x+B2y+C2=0A3x+B3y+C3=0 \begin{aligned} A_1x + B_1y + C_1 &= 0 \\ A_2x + B_2y + C_2 &= 0 \\ A_3x + B_3y + C_3 &= 0 \end{aligned}

is given by

A=A1B1C1A2B2C2A3B3C322A1B1A2B2A2B2A3B3A3B3A1B1 A = \frac{\begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix}^2} {2\cdot \begin{vmatrix} A_1 & B_1 \\ A_2 & B_2 \end{vmatrix} \cdot \begin{vmatrix} A_2 & B_2 \\ A_3 & B_3 \end{vmatrix} \cdot \begin{vmatrix} A_3 & B_3 \\ A_1 & B_1 \end{vmatrix}}

The area of a triangle whose vertices are P1(x1,y1),P2(x2,y2)P_1(x_1, y_1) , P_2(x_2, y_2)and P3(x3,y3)P_3(x_3, y_3) is given by :

A=12x1y11x2y21x3y31 A = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}

and by:

A=12x2x1y2y1x3x1y3y1 A = \frac{1}{2} \begin{vmatrix} x_2-x_1 & y_2-y_1 \\ x_3-x_1 & y_3-y_1 \end{vmatrix}

Centroid

The centroid of a triangle whose vertices are P1(x1,y1),P2(x2,y2)P_1(x_1,y_1), P_2(x_2, y_2) and P3(x3,y3)P_3(x_3, y_3) is given by:

(x,y)=(x1+x2+x33,y1+y2+y33)(x,y) = \left( \frac{x_1+x_2+x_3}{3} , \frac{y_1+y_2+y_3}{3} \right)

Incenter

The incenter of a triangle whose vertices are P1(x1,y1),P2(x2,y2)P_1(x_1,y_1), P_2(x_2, y_2) and P3(x3,y3)P_3(x_3, y_3) is given by:

(x,y)=(ax1+bx2+cx33,ay1+by2+cy33)(x,y) = \left( \frac{a\,x_1+b\,x_2+c\,x_3}{3} , \frac{a\,y_1+b\,y_2+c\,y_3}{3} \right)

where aa is the length of P2P3P_2P_3, bb is the length of P3P1P_3P_1, and cc is the length of P1P2P_1P_2.

Circumcenter

The circumcenter of a triangle whose vertices are P1(x1,y1),P2(x2,y2)P_1(x_1,y_1), P_2(x_2, y_2) and P3(x3,y3)P_3(x_3, y_3) is given by:

(x,y)=( x12+y12y11x22+y22y21x32+y32y312x1y11x2y21x3y31 , x1x12+y121x2x22+y221x3x32+y3212x1y11x2y21x3y31 ) (x , y) = \left( ~ \frac{\begin{vmatrix} x_1^2+y_1^2 & y_1 & 1 \\ x_2^2+y_2^2 & y_2 & 1 \\ x_3^2+y_3^2 & y_3 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~,~ \frac{\begin{vmatrix} x_1 & x_1^2+y_1^2 & 1 \\ x_2 & x_2^2+y_2^2 & 1 \\ x_3 & x_3^2+y_3^2 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~ \right)

Orthocenter

The orthocenter of a triangle whose vertices are P1(x1,y1),P2(x2,y2)P_1(x_1,y_1), P_2(x_2, y_2) and P3(x3,y3)P_3(x_3, y_3) is given by:

(x,y)=( y1x2x3+y121y2x3x1+y221y3x1x2+y3212x1y11x2y21x3y31 , x12+y2y3x11x22+y3y1x21x32+y1y2x312x1y11x2y21x3y31 ) (x , y) = \left( ~ \frac{\begin{vmatrix} y_1 & x_2x_3+y_1^2 & 1 \\ y_2 & x_3x_1 + y_2^2 & 1 \\ y_3 & x_1x_2+y_3^2 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~,~ \frac{\begin{vmatrix} x_1^2+y_2y_3 & x_1 & 1 \\ x_2^2+y_3y_1 & x_2 & 1 \\ x_3^2+y_1y_2 & x_3 & 1 \\ \end{vmatrix}} {2 \cdot \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}}~ \right)

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