If the areas of the 4 faces of a tetrahedron are all equal, prove that it's a regular tetrahedron?

Question

Assuming that the tetrahedron has a non-zero volume.

mikedotcom, might not it be possible to have an irregular tetrahedron with 4 faces all of equal areas?

My bad, tinker, I missed this part when working on a related problem. Thanks for having pointed this one out to me. This is now a non-problem.

Answer 1

This sounds a lot like one of Alexanders questions. One can't prove that the tetrahedron is regular just from knowing the areas of the sides are equal, any four identical isosceles triangles, equilateral or not can be assembled into a tetrahedron.

Answer 2

Revised for readability: permit a, b, c, d ? ?² be the respective projections onto the 1st 2 coordinates of the vertices ?, ?, ?, ? of a regularly occurring tetrahedron sitting in ?³. be conscious this projection is orthogonal. by technique of translating by technique of their mean, we are in a position to anticipate ? + ? + ? + ? = (0,0,0). It follows that a+b+c+d = (0,0). Denote their negatives as -a = a', -b = b', -c = c', -d = d', that are projections of -?, -?, -?, -?, respectively. Now all 8 factors mutually are the orthogonally projected vertices of a cube sitting in ?³. we are in a position to particular those 8 factors as all ± combos ±u ± v ±w the place u = (a-b')/2 = (a+b)/2, v = (a-c')/2 = (a+c)/2, w = (a-d')/2 = (a+d)/2. be conscious that u, v, and w are projections of (? + ?)/2, (? + ?)/2, and (? + ?)/2, that are together orthogonal and each and all of the same length (Use ? + ? + ? + ? = (0,0,0)). permit D be the three-by technique of-3 matrix whose rows are those row vectors (in that order). be conscious D is an orthogonal matrix (interior the experience that DD? is a scalar distinctive of I). Now permit C be the three-by technique of-2 matrix whose rows are the row vectors u, v, and w, in that order. be conscious C is in basic terms the 1st 2 columns of D. It follows that the columns of C are orthogonal (as column vectors) and function the same sq. modulus. that's no longer difficult to be sure that this necessary subject (that the columns of C are orthogonal as column vectors and function the same sq. modulus) is likewise a sufficient subject for there to exist a regularly occurring tetrahedron of whose vertices a, b, c, and d are the projections: certainly, the 0.33 column of D could nicely be taken to be the normalized pass made of the columns of C. The vectors a, b, c, and d could nicely be recovered from the rows of C with the aid of fact the combos ±u ± v ± w having an excellent extensive style (0 or 2) of minus indicators.

Answer 3

A regular tetrahedron is one in which the four triangles are regular, or "equilateral"
If all six edges of a tetrahedron are equal, it is called a regular tetrahedron.

Answer 4

You can become a top contributor by just copying stuff from Wikipedia, even if it doesnt answer the question??? (a proof is asked for)