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A Quartet of Tetrahedra
This feature is all about tetrahedra, with four problems for you to solve.
You can find another problem involving a tetrahedron in STEP Support Programme Foundation Assignment 5.
Tetra Inequalities
Age 16 to 18
Challenge Level 
Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?
Tetra Slice
Age 16 to 18
Challenge Level
Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?
Tetra Perp
Age 16 to 18
Challenge Level
Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.
Pythagoras for a Tetrahedron
Age 16 to 18
Challenge Level
In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.
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