Right Triangle Tetrahedron

Geometry Level 3

True or False: There exists a tetrahedron where all four faces are right triangles and all six sides are integers.

True False

1 solution

David Vreken
Oct 23, 2019

One example is tetrahedron $ABCD$ where $AB = 680$ , $AC = 697$ , $AD = 672$ , $BC = 153$ , $BD = 104$ , and $CD = 185$ .

Then for the faces:

$\triangle ABC$ : $153^2 + 680^2 = 697^2$

$\triangle ABD$ : $104^2 + 672^2 = 680^2$

$\triangle ACD$ : $185^2 + 672^2 = 697^2$

$\triangle BCD$ : $104^2 + 153^2 = 185^2$

so that all the faces are right triangles with integer sides.

How would you solve the question without knowing this?

Elijah L - 1 year, 7 months ago

I ultimately used a computer program to help me, but you can narrow down the search by proving that the only tetrahedron possible is one where two of the right angles meet at one vertex and two of the other right angles meet at another vertex.

David Vreken - 1 year, 7 months ago

http://mathworld.wolfram.com/PerfectCuboid.html

Yuriy Kazakov - 1 year, 7 months ago

That's slightly different, as a tetrahedron made from a perfect cuboid would have a fourth face (made from the diagonals) that cannot be a right triangle. It is interesting that Euler's perfect cuboid had many of the same numbers as the tetrahedron that I provided in my solution, though.

David Vreken - 1 year, 7 months ago

Try this link

Mark Hennings - 1 year, 7 months ago

Right Triangle Tetrahedron

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