Linear Interpolation

I have a 3-D traingle, and the edges are a,b,c. If I want to find a line interpolation at a point in the center, let's say that it's P(x,y)....

My equations are
[tex]\Delta Z[/tex] = A + Bx + Cy
[tex]\Delta Z_a[/tex] = A + Bx_a + Cy_a
[tex]\Delta Z_b[/tex] = A + Bx_b + Cy_b
[tex]\Delta Z_c[/tex] = A + Bx_c + Cy_c

In order to solve for [tex]\Delta Z[/tex], how do I use the above equations? Do I have to add them (equations 2,3,4) all up and substitute in A for the first equation?

To find the coefficients, do I just solve for A,B,C after I know what [tex]\Delta Z[/tex] is?

I also have to find out P(y) by assuming that I know what P(x,z) is. For this do I just substiture y for z to the same equations?

Also, I have to find a best average planar rectangle from this.

I'd appreciate any help.
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