Theorem : The area of a triangle the coordinates of whose vertices are `(x_1;y_1);(x_2;y_2)and (x_3;y_3)` is 1/2|(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|`

Theorem: Prove that the coordinates of centroid of the triangle whose coordinates are (x_(1);y_(1));(x_(2);y_(2)) and (x_(3);y_(3)) are ((x_(1)+x_(2)+x_(3))/(3);(y_(1)+y_(2)+y_(3))/(3))

Write the formula for the area of the triangle having its vertices at (x_(1),y_(1)),(x_(2),y_(2)) and (x_(3),y_(3))

A triangle has its three sides equal to a , b and c . If the coordinates of its vertices are A(x_1, y_1),B(x_2,y_2)a n dC(x_3,y_3), show that |[x_1,y_1, 2],[x_2,y_2, 2],[x_3,y_3, 2]|^2=(a+b+c)(b+c-a)(c+a-b)(a+b-c)

The centroid of the triangle with vertices at A(x_(1),y_(1)),B(x_(2),y_(2)),c(x_(3),y_(3)) is

Find the coordinates of the centroid of a triangle having vertices P(x_1,y_1,z_1),Q(x_2,y_2,z_2) and R(x_3,y_3,z_3)

STATEMENT-1 : The centroid of a tetrahedron with vertices (0, 0,0), (4, 0, 0), (0, -8, 0), (0, 0, 12)is (1, -2, 3). and STATEMENT-2 : The centroid of a triangle with vertices (x_(1), y_(1), z_(1)), (x_(2), y_(2), z_(2)) and (x_(3), y_(3), z_(3)) is ((x_(1)+x_(2)+x_(3))/3, (y_(1)+y_(2)+y_(3))/3, (z_(1)+z_(2)+z_(3))/3)

If the co-ordinates of the vertices of an equilateral triangle with sides of length a are (x_1,y_1), (x_2, y_2), (x_3, y_3), then |[x_1,y_1,1],[x_2,y_2,1],[x_3,y_3,1]|=(3a^4)/4

If the coordinates of the vertices of an equilateral triangle with sides of length a are (x_(1),y_(1)),(x_(2),y_(2)) and (x_(3),y_(3)) then |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^2=(3a^(4))/4