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 d \ C(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)`

A triangle has its three sides equal to a , ba n dc . 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)

An equilateral triangle has each side to a. If the coordinates of its vertices are (x_(1), y_(1)), (x_(2), y_(2)) and (x_(3), y_(3)) then the square of the determinat |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)| equals

If A(x_1, y_1),B(x_2, y_2) and C(x_3,y_3) are vertices of an equilateral triangle whose each side is equal to a , then prove that |[x_1,y_1, 2],[x_2,y_2, 2],[x_3,y_3, 2]|^2=3a^4

An equilateral triangle has each of its sides of length 6 cm. If (x_1, y_1);(x_2, y_2)&(x_3, y_3) are its vertices, then the value of the determinant |x_1y_1 1x_2y_2 1x_3y_3 1|^2 is equal to: 192 (b) 243 (c) 486 (d) 972

An equilateral triangle has each of its sides of length 6 cm. If (x_1, y_1);(x_2, y_2)&(x_3, y_3) are its vertices, then the value of the determinant |x_1y_1 1x_2y_2 1x_3y_3 1|^2 is equal to: 192 (b) 243 (c) 486 (d) 972

If (x_1-x_2)^2+(y_1-y_2)^2=144 ,(x_2-x_3)^2+(y_2-y_3)^2=25 and (x_3-x_1)^2+(y_3-y_1)^2=169 , then the value of |[x_1,y_1, 1],[x_2,y_2 ,1],[x_3,y_3, 1]|^2 is 30 (b) 30^2 (c) 60 (d) 60^2

If |x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_3 1| then the two triangles with vertices (x_1, y_1),(x_2,y_2),(x_3,y_3) and (a_1,b_1),(a_2,b_2),(a_3,b_3) are equal to area (b) similar congruent (d) none of these

If |x_1y_1 1x_2y_2 1x_3y_3 1|=|a_1b_1 1a_2b_2 1a_3b_3 1| then the two triangles with vertices (x_1, y_1),(x_2,y_2),(x_3,y_3) and (a_1,b_1),(a_2,b_2),(a_3,b_3) are equal to area (b) similar congruent (d) none of these

Show that the plane a x+b y+c z+d=0 divides the line joining the points (x_1, y_1, z_1)a n d(x_2, y_2, z_2) in the ratio (a x_1+b y_1+c z_1+d)/(a x_2+b y_2+c z_2+d) .

If A(x_(1), y_(1)), B(x_(2), y_(2)) and C (x_(3), y_(3)) are the vertices of a Delta ABC and (x, y) be a point on the internal bisector of angle A, then prove that b|(x,y,1),(x_(1),y_(1),1),(x_(2),y_(2),1)|+c|(x,y,1),(x_(1),y_(1),1),(x_(3),y_(3),1)|=0 where, AC = b and AB = c.