This system of nonlinear equations was proposed by an artificial intelligence named Alice.

f1 := x4^3+x5^2*x6-sin(x3)+exp(x1*x2)-5; 
f2 := ln(x4+x5)-x6^2+x3^3-x1^2*x2-2; 
f3 := cos(x4*x5)+x6*sin(x3)-x1*x2^2+x4^2*x5^3;

At the same time, AI accompanied it with comments. Here are some of them:
 

### System Characteristics
 **Underdetermination**: 6 unknowns with 3 equations 

### Solution Complexity

* Lack of a general analytical solution
* Need to use numerical methods
* Possibility of multiple solutions
* Complexity of visualization
* Sensitivity to initial approximations

### Solution Methods

* **Iterative methods**:
* Newton's method
* Simple iteration method
* Gradient methods
* **Numerical methods**:
* Finite difference method
* Monte Carlo method
* Genetic algorithms
* **Optimization approaches**:
* Lagrange multiplier method
* Constrained optimization methods

### Practical Application

Similar systems are found in:
* Quantum mechanics
* Field theory
* Economic modeling
* Bioinformatics
* Machine learning
* Engineering calculations of complex systems

Analyzing such systems often requires specialized software and powerful computing resources.

It is possible to construct a system of equations that defines all spheres that touch two smooth surfaces.

This includes all the spheres inscribed between these surfaces. The first two equations are surfaces,  x7, x8, x9 are the coordinates of the centers of the spheres, the remaining variables correspond to the coordinates of the points of contact. 
In the program, we inscribe a sphere between a cylinder and a sphere. An example of obtaining one of the infinite subsets of the solution to a system of equations is implemented. This solution is "responsible" for the equation f8.
INSCRIBED_SPHERES_CILL_FOR.mw

And one more example.

Geometrical entertainment in the form of rolling without slipping, now inside a torus.
You can also print out the corresponding equations - these graphs are # in the text, but it is better to do this separately from the geometric animation, because textplot3d takes up a lot of resources.

Please consider it not as a Maple program, but simply as an idea for a corresponding algorithm.
for_TORUS_IN_TORUS_for.mw
 

An example of solving a system of nonlinear equations, when all (real) solutions are found for one initial approximation. To make it easier to understand the program text, a description of the solution method is provided separately in PDF format.

Draghilev’s_method_pdf.pdf

3_EQUATIONS.mw

Using the curve, the projection of which from the space R^4 to R^3 we see in the figure, we find good approximations to the solution. For each solution, the program prints out the norm of the discrepancy of the entire system of equations and shows a simple graph when the additional variable intersects R^3 . At this moment of intersection, the solution of our system is fixed.
System of equations and initial approximation.
 

f1 := x1^2+x2^2-x3^2; 
f2 := (x1-1)^4+x2^4+(x3-2)^4-81; 
f3 := x3-3*tan(.4*x2); 
x01, x02, x03 := 1, 1, 1

Old link to a similar example, where the dimension of the system of equations is 2, and we can see the auxiliary curve completely in R^3.

This is a task from one forum:  “Let's mark an arbitrary point on the circle. Let's draw a segment from this point, perpendicular to the diameter, and draw a circle, the center of which is at this point, and the radius is equal to this segment. Let's mark the intersection point of the segment connecting the intersection points of the circles with the perpendicular segment. Prove that the locus of all such points is an ellipse.”
I wanted to get a picture of a numerically animated "proof" using Maple tools.
МАTH_HЕLP_PLANET.mw
 And in fact, it turned out that AB=2AC, or AC=BC.