Find set of equations for N+1 hyperplanes in N dimensions such that no hyperplane is parallel and all hyperplanes are equidistant from origin.
Note: You must start with a random hyperplane in N dimension, and then rotate around origin such that it creates a new hyperplane. Once, you have rotated that intial hyperplane in same manner for a N+2 times, you must get the equation for same hyperplane that you started with.
Example: For 2 dimensions, take a line and now rotate it about origin for 120 degrees, do this twice, and you get a equilateral triangle with its centre at the origin. For N=3, you have to find the equations of the 4 planes making the up the tetrahedral volume with its centre at origin. Now, simple do this for N dimensions, where each hyperplane in Nth dimension is composed of N points, form all equations with respect to these points.
Take the standard basis of $\mathbb{R}^{n+1}$, $$ e_0, e_1, \cdots, e_n. $$ Choose $n$-many vectors out of these. For example, you may choose $$e_1, \cdots, e_n.$$ Write down the equation for the $n$-dimensional hyperplane passing through these vectors, which is $$ \sum_{j=1}^n t_j e_j =1, \sum_{j=1}^n t_j=1. $$ You can do the same thing with other choices, and they are not parrellel to each other, and the distances from those to the origin are all equal.
If you want to start with the random $N$-dimensional hyperplane, just rotate and move it to identify with one of those plane passing through $N$ vectors among the standard basis, write down the equation for other hyperplanes, and rotate and move them in the opposite way you did at the beginning.