The link below shows the vertex results using wolfram cloud, which is much faster.

# How do I solve for the coordinates of the vertices of a Trirectangular Tetrahedron using the symbolic toolbox?

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The coordinates of the three points on the base of a right-angled tetrahedron ABCD (Trirectangular Tetrahedron: a tetrahedron with three prisms perpendicular to each other at a common vertex) are known to be B(x1,y1,z1), C(x2,y2,z2) and D(x3,y3,z3). vertex A is at right angles. Take only one of the symmetries,how to find the coordinates of the vertex A?The schematic diagram is as follows.

已知直角四面体ABCD(共顶点的三条棱线互相垂直的四面体)的底面三个点的坐标分别为B(x1,y1,z1),C(x2,y2,z2),D(x3,y3,z3),求顶点A的坐标？示意图如下：

How to find the coordinates of the point A through the symbolic toolbox, I tried to solve it but the calculation took a long time and I still could not get an analytical solution, even though I made many assumptions.

syms x y z x1 y1 z1 x2 y2 z2 x3 y3 z3

assume([x y z x1 y1 z1 x2 y2 z2 x3 y3 z3],'real')

pointA = [x,y,z];

pointB = [x1,y1,z1];

pointC = [x2,y2,z2];

pointD = [x3,y3,z3];

AB = pointB-pointA;

AC = pointC-pointA;

AD = pointD-pointA;

BC = pointC- pointB;

BD = pointD- pointB;

CD = pointD- pointC;

assumeAlso(dot(BC,BD)>0 & dot(-BD,-CD)>0 & dot(-BC,CD)>0) % triangular BCD is an acute triangle,make sure you can find the solution set

equ1 = dot(AB,AC)==0;

equ2 = dot(AC,AD)==0;

equ3 = dot(AB,AD)==0;

sol2 = solve([equ1,equ2,equ3],[x,y,z])

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