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

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)

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) .

ax+by=1 bx+ay=2

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 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.

det[[ Prove that: ,b,ax+bya,c,bx+cyax+by,bx+cy,0]]=(b^(2)-ac)(ax^(2)+2bxy+cy^(2))

If x + y + z = 0, prove that |(ax,by,cz),(cy,az,bx),(bz,cx,ay)|=xyz|(a,b,c),(c,a,b),(b,c,a)|

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

prove that |{:((a-x)^(2),,(a-y)^(2),,(a-z)^(2)),((b-x)^(2),,(b-y)^(2),,(b-z)^(2)),((c-x)^(2),,(c-y)^(2),,(c-z)^(2)):}| |{:((1+ax)^(2),,(1+bx)^(2),,(1+cx)^(2)),((1+ay)^(2),,(1+by)^(2),,(1+cy)^(2)),((1+az)^(2),,(1+bx)^(2),,(1+cz)^(2)):}| =2 (b-c)(c-a)(a-b)xx (y-z) (z-x)(x-y)