How to proof that two lines in cube are perpendicular, without use of vectors Given: Cube $ABCDA_1B_1C_1D_1$
Prove that $BD$ is perpendicular to $AC_1$

I don't have any idea how to proof this. Also I can't use vectors(we didn't study them in school). I can use all theorems from the stereometry(I think another name for this is solid geometry, but basically we deal with 3d figures(finding their volume, area, angles between different sides etc..), planes, and lines in the space)

Without calculation. Just use axioms.

*

*$AC$ is perpendicular to $BD$.

*$A'C'$ is perpendicular to $B'D'$.

There is an axiom saying that

There is a single plane passing through two parallel lines.


*

*The plane $ACC'A'$ is perpendicular to $BDD'B'$.

Another axioms:

If a line $L$ is perpendicular to a plane then every line on the plane is perpendicular to $L$.


*

*$BD$ is perpendicular to plane $ACC'A'$ so the line $AC'$ on the plane is also perpendicular to $BD$.


*Done!
