Folklore

In the tetrahedron $A B C D$, the dihedral angles $B A D$ and $B C D$ are obtuse. Compare the lengths of the edges $A C$ and $B D$.
The first method. Construct a sphere with diameter $B D$. Since angles $B A D$ and $B C D$ are obtuse, vertices $A$ and $C$ will lie inside this sphere. Consequently, the distance from $A$ to $C$ is less than the diameter.

The second method. In triangles $A B D$ and $C B D$, draw medians $A M$ and $C M$ respectively (see figure).

![](https://cdn.mathpix.com/cropped/2024_05_06_60e93adedf73503f106cg-31.jpg?height=643&width=592&top_left_y=1628&top_left_x=733)

In an obtuse triangle, the median to the larger side is less than half of that side. Therefore, $A C < A M + C M < B D$.

## Answer

$A C < B D$.