# Approximating Line Intersection Equation with Maple

## Main Question or Discussion Point

I'm having a hard time creating a tidy equation to solve this seemingly simple problem.

I have two points in standard 3D cordinate space, we'll call them C and D to be consistent with my work. The location of these points is given by (Ci,Cj,Ck) and (Di,Dj,Dk).

I then have a third point, A, given by (Ai, Aj, (Ak-x)). The distance between point C and A is equal to constC, and the distance between D and A is equal to constD.

Given C, D, constC, constD, x, and Ak, I need to solve for Ai and Aj.

Well the first thing I did was create two equations that set the magnitude of AC and AD equal to constC and constD. Using Maple to solve for Ai and Aj gives me an entire screen of numbers full of RootOf expressions. If I give Maple some real numbers for the given variables then it quickly outputs the two possible solutions for Ai and Aj, which I've verified to be correct.

Here is my Maple code.

C_i:=C_i:
Cj:=Cj:
Ck:=Ck:
Di:=Di:
Dj:=Dj:
Dk:=Dk:
Ak:=Ak:
x:=x:
constC:=constC:
constD:=constD:

solve({(sqrt((Ai-C_i)^2+(Aj-Cj)^2+((Ak+x)-Ck)^2))=constC,(sqrt((Ai-Di)^2+(Aj-Dj)^2+((Ak+x)-Dk)^2))=constD},{Ai,Aj});

I'm looking for a compact equation (that I can insert into my excel spreadsheet) that approximates Ai and Aj. I am still learning Maple and my math skills are somewhat limited. Any help would be appreciated.

Thanks,
- Steven

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