Prove that `|[x_1, y_1, 1] , [ax_1+bx_2+cx_3, ay_1+by_2+cy_3, a+b+c] , [-ax_1+bx_2+cx_3, -ay_1+by_2+cy_3, -a+b+c]|=0`

Prove that |{:(x_1,y_1,1),(ax_1+bx_2+cx_3,ay_1+by_2+cy_3,a+b+c),(-ax_1+bx_2+cx_3,-ay_1+by_2+cy_2,-a+b+c):}|=0

Prove that |{:(x_1,y_1,1),(ax_1+bx_2+cx_3,ay_1+by_2+cy_3,a+b+c),(-ax_1+bx_2+cx_3,-ay_1+by_2+cy_2,-a+b+c):}|=0

Prove that: |[a, b, ax+by],[ b, c, bx+cy], [ax+by, bx+cy,0]|=(b^2-a c)(a x^2+2b x y+c y^2)

Prove that the coordinates of the centre of the circle inscribed in the triangle,whose vertices are the points (x_(1),y_(1)),(x_(2),y_(2)) and (x_(3),y_(3)) are (ax_(1)+bx_(2)+cx_(3))/(a+b+c) and (ay_(1)+by_(2)+cy_(3))/(a+b+c)

Show that the plane ax+by+cz+d=0 divides the line joining (x_1, y_1, z_1) and (x_2, y_2, z_2) in the ratio of (-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))

Show that the plane ax+by+cz+d=0 divides the line joining (x_1, y_1, z_1) and (x_2, y_2, z_2) in the ratio of (-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))

Show that the plane ax+by+cz+d=0 divides the line joining (x_1, y_1, z_1) and (x_2, y_2, z_2) in the ratio of (-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))

Show that the plane ax+by+cz+d=0 divides the line joining (x_1, y_1, z_1) and (x_2, y_2, z_2) in the ratio of (-(ax_1+ay_1+cz_1+d)/(ax_2+by_2+cz_2+d))

Prove that |(a,b,ax+by),(b,c,bx+cy),(ax+by, bx + cy, 0)| = (b^(2)-ac)(ax^(2) + 2bxy + cy^(2)) .

STATEMENT-1 : If a, b, c are distinct and x, y, z are not all zero and ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 , then a + b + c = 0 and STATEMENT-2 : a^2 + b^2 + c^2 > ab + bc + ca , if a, b, c are distinct.