3D geometry, what are the coordinates of the 4th vertex and the point of intersection of this trapezoid?

Question

3 Vertex of the trapezoid are given : A(4,-1,2) B(7,1,-3) D(0,-4,6) and we know that AB and CD are parallel, and CD=2AB (opposite vertices are B-D and A-C)

The question is : what are the coordinates of vertex C, and what are the coordinates of the point of intersection of the diagonals?

Thank you!

Answer 1

Hint: From parallelism, $\vec{AB} = k\vec{CD}$ for some negative $k$. Find $k$; so $1/k(B-A) + D = C$. To get the intersection of the diagonals, look at the lines $t(C - A)$ and $r(D-B)$ and their intersection.

Answer 2

Let $$C[x_C,y_C,z_c]$$ then $$2[0-x_C,-4-y_C,6-z_C]=[7-4,1+1,-3-2]$$ Can you solve this? You will get $$-2x_C=3,-8-2y_C=2,12-2z_C=-5$$