Coordinates of Triangle Centers

Vertices: $$A(x_1,y_1),\quad B(x_2,y_2),\quad C(x_3,y_3)$$; sides: $$a=BC,\;b=CA,\;c=AB$$.

Centroid $$G$$: $$\Bigl(\frac{x_1+x_2+x_3}{3},\;\frac{y_1+y_2+y_3}{3}\Bigr)$$.

Incenter $$I$$: $$\Bigl(\frac{a x_1 + b x_2 + c x_3}{a+b+c},\;\frac{a y_1 + b y_2 + c y_3}{a+b+c}\Bigr)$$.

Circumcenter $$O$$: Let $$D = 2\bigl[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\bigr].$$
Then$$\displaystyle O=\Bigl(\frac{(x_1^2+y_1^2)(y_2-y_3)+(x_2^2+y_2^2)(y_3-y_1)+(x_3^2+y_3^2)(y_1-y_2)}{D},\;\frac{(x_1^2+y_1^2)(x_3-x_2)+(x_2^2+y_2^2)(x_1-x_3)+(x_3^2+y_3^2)(x_2-x_1)}{D}\Bigr).$$

Orthocenter $$H$$: $$H = A+B+C - 2O$$ as vectors, i.e., $$H =\bigl(x_1+x_2+x_3 -2x_O,\;y_1+y_2+y_3 -2y_O\bigr).$$