已知 ae , be , ce , a.x , a.y , a.z, b.x, b.y, b.z , c.x , c.y, c.z, 求解(x - a.x)^2 + (y - a.y)^2 + (z - a.z)^2 - ae * ae;
(x - b.x)^2 + (y - b.y)^2 + (z - b.z)^2 - be * be;
(x - c.x)^2 + (y - c.y)^2 + (z - c.z)^2 - ce * ce; 方程组，并将x,y,z以最终方程式展示出来

我们有三个方程，分别为：

\[
(x - a_x)^2 + (y - a_y)^2 + (z - a_z)^2 - ae^2 = 0
\]
\[
(x - b_x)^2 + (y - b_y)^2 + (z - b_z)^2 - be^2 = 0
\]
\[
(x - c_x)^2 + (y - c_y)^2 + (z - c_z)^2 - ce^2 = 0
\]

展开这些方程，并简化每一个方程，得到：

### 展开第一个方程
\[
(x - a_x)^2 + (y - a_y)^2 + (z - a_z)^2 - ae^2 = 0
\]

\[
(x^2 - 2a_xx + a_x^2) + (y^2 - 2a_yy + a_y^2) + (z^2 - 2a_zz + a_z^2) - ae^2 = 0
\]

\[
x^2 + y^2 + z^2 - 2a_xx - 2a_yy - 2a_zz + (a_x^2 + a_y^2 + a_z^2 - ae^2) = 0
\]

### 展开第二个方程
\[
(x - b_x)^2 + (y - b_y)^2 + (z - b_z)^2 - be^2 = 0
\]

\[
(x^2 - 2b_xx + b_x^2) + (y^2 - 2b_yy + b_y^2) + (z^2 - 2b_zz + b_z^2) - be^2 =

Explore how to solve equations involving spheres and distances with step-by-step explanations. This problem involves applying the equation of a sphere and the distance formula to find x, y, and z coordinates. Suitable for advanced mathematics students.

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