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

det[[ Prove that x_(1),y_(1),1ax_(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

ax+by=1 bx+ay=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))

[[a, b, ax + byb, c, bx + cyax + by, bx + cy, 0]] = (b ^ (2) -ac) (ax ^ (2) + 2bxy + cy ^ (2))

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 lines x=ay +b,z =cy +d and x=a'y +b' z =c'y +a' are perpendicular if aa'+cc' +1=0

Prove that the incentre of the triangle whose vertices are given by A(x1,y1),B(x2,y2),C(x3,y3)is(ax1+bx2+cx3)/(a+b+c),(ay1+by2+cy2)/(a+b+c) where a b,and c are the sides opposite to the angles A,B and C respectively.