Area of the triangle

The area of the triangle formed by the three lines:

A1x+B1y+C1A2x+B2y+C2A3x+B3y+C3=0=0=0

is given by

A=∣∣∣∣A1A2A3B1B2B3C1C2C3∣∣∣∣22⋅∣∣∣A1A2B1B2∣∣∣⋅∣∣∣A2A3B2B3∣∣∣⋅∣∣∣A3A1B3B1∣∣∣

The area of a triangle whose vertices are P1(x1,y1),P2(x2,y2)and P3(x3,y3) is given by :

A=12∣∣∣∣x1x2x3y1y2y3111∣∣∣∣

and by:

A=12∣∣∣x2−x1x3−x1y2−y1y3−y1∣∣∣

Centroid

The centroid of a triangle whose vertices are P1(x1,y1),P2(x2,y2) and P3(x3,y3) is given by:

(x,y)=(x1+x2+x33,y1+y2+y33)

Incenter

The incenter of a triangle whose vertices are P1(x1,y1),P2(x2,y2) and P3(x3,y3) is given by:

(x,y)=(ax1+bx2+cx33,ay1+by2+cy33)

where a is the length of P2P3, b is the length of P3P1, and c is the length of P1P2.

Circumcenter

The circumcenter of a triangle whose vertices are P1(x1,y1),P2(x2,y2) and P3(x3,y3) is given by:

(x,y)=⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ∣∣∣∣∣x21+y21x22+y22x23+y23y1y2y3111∣∣∣∣∣2⋅∣∣∣∣x1x2x3y1y2y3111∣∣∣∣ , ∣∣∣∣∣x1x2x3x21+y21x22+y22x23+y23111∣∣∣∣∣2⋅∣∣∣∣x1x2x3y1y2y3111∣∣∣∣ ⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

Orthocenter

The orthocenter of a triangle whose vertices are P1(x1,y1),P2(x2,y2) and P3(x3,y3) is given by:

(x,y)=⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜ ∣∣∣∣∣y1y2y3x2x3+y21x3x1+y22x1x2+y23111∣∣∣∣∣2⋅∣∣∣∣x1x2x3y1y2y3111∣∣∣∣ , ∣∣∣∣∣x21+y2y3x22+y3y1x23+y1y2x1x2x3111∣∣∣∣∣2⋅∣∣∣∣x1x2x3y1y2y3111∣∣∣∣ ⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟