((x00 - y00) ^ 2 + (x01 - y01) ^ 2 + (x10 - y10) ^ 2 + (x11 - y11) ^ 2) *((y00 - z00) ^ 2 + (y01 - z01) ^ 2 + (y10 - z10) ^ 2 + (y11 - z11) ^ 2) -(   (x00 - z00) ^ 2 + (x01 - z01) ^ 2 + (x10 - z10) ^ 2 + (x11 - z11) ^ 2 -((x00 - y00) ^ 2 + (x01 - y01) ^ 2 + (x10 - y10) ^ 2 + (x11 - y11) ^ 2 +((y00 - z00) ^ 2 + (y01 - z01) ^ 2 + (y10 - z10) ^ 2 + (y11 - z11) ^ 2))) ^ 2 - (((y00 - x00) * (y00 - z00) + (y01 - x01) * (y01 - z01) + (y10 - x10) * (y10 - z10) + (y11 - x11) * (y11 - z11)) ^ 2 + ((x00 - y00) ^ 2 + (x01 - y01) ^ 2 + (x10 - y10) ^ 2 + (x11 - y11) ^ 2) * ((y00 - z00) ^ 2 + (y01 - z01) ^ 2 + (y10 - z10) ^ 2 + (y11 - z11) ^ 2)) 

The expression you've written is quite complex, with many terms involving squares and products of differences between coordinates. Let’s break it down for clarity:

The general structure of your equation looks like this:

\[
\left( (x_{00} - y_{00})^2 + (x_{01} - y_{01})^2 + (x_{10} - y_{10})^2 + (x_{11} - y_{11})^2 \right) \cdot \left( (y_{00} - z_{00})^2 + (y_{01} - z_{01})^2 + (y_{10} - z_{10})^2 + (y_{11} - z_{11})^2 \right)
\]
\[
-\left( (x_{00} - z_{00})^2 + (x_{01} - z_{01})^2 + (x_{10} - z_{10})^2 + (x_{11} - z_{11})^2 - \left( (x_{00} - y_{00})^2 + (x_{01} - y_{01})^2 + (x_{10} - y_{10})^2 + (x_{11} - y_{11})^2 + \left( (y_{00} - z_{00})^2 + (y_{01} - z_{01})^2 + (y_{10} - z_{10})^2 + (y_{11} - z_{11})^2 \right)\right)\right)^2
\]
\[
-\left( \left( (y_{00} - x_{00})(y_{00} - z_{00}) + (y_{01} - x_{01})(y_{01} - z_{01}) + (y_{10} - x_{10})(y_{10} - z_{10}) + (y_{11} - x_{11})(y_{11} - z_{11}) \right)^2
+ \left( (x_{00} - y_{00})^2 + (x_{01} - y_{01})^2 + (x_{10} - y_{10})^2 + (x_{11} - y_{11})^2 \right) \cdot \left( (y_{00} - z_{00})^2 + (y_{01} - z_{01})^2 + (y_{10} - z_{10})^2 + (y_{11} - z_{11})^2 \right) \right)
\]

### Steps to Simplify:

1. **Group like terms:** Identify common factors to simplify parts of the expression where squares and sums are repeated.
2. **Expand products and powers:** If needed, expand the terms involving products of sums to simplify.
3. **Combine similar terms:** After expansion, combine the terms where possible.
4. **Check for factoring opportunities:** You might notice patterns that allow factoring to simplify the expression further.

Would you like me to continue with a detailed step-by-step simplification?

((x00 - y00) ^ 2 + (x01 - y01) ^ 2 + (x10 - y10) ^ 2 + (x11 - y11) ^ 2) *((y00 - z00) ^ 2 + (y01 - z01) ^ 2 + (y10 - z10) ^ 2 + (y11 - z11) ^ 2) -(   (x00 - z00) ^ 2 + (x01 - z01) ^ 2 + (x10 - z10) ^ 2 + (x11 - z11) ^ 2 -((x00 - y00) ^ 2 + (x01 - y01) ^ 2 + (x10 - y10) ^ 2 + (x11 - y11) ^ 2 +((y00 - z00) ^ 2 + (y01 - z01) ^ 2 + (y10 - z10) ^ 2 + (y11 - z11) ^ 2))) ^ 2 - (((y00 - x00) * (y00 - z00) + (y01 - x01) * (y01 - z01) + (y10 - x10) * (y10 - z10) + (y11 - x11) * (y11 - z11)) ^ 2 + ((x00 - y00) ^ 2 + (x01 - y01) ^ 2 + (x10 - y10) ^ 2 + (x11 - y11) ^ 2) * ((y00 - z00) ^ 2 + (y01 - z01) ^ 2 + (y10 - z10) ^ 2 + (y11 - z11) ^ 2))

Explore the solution to a complex algebraic expression involving multiple squared differences and vector dot products. This detailed breakdown simplifies the equation using algebraic and geometric principles. Suitable for advanced high school or undergraduate students.

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