# Pyramid with unit sides inside a cube

Let $ABCDEFGH$ be a unit cube with base $ABCD$. Let $P$ be the top of the pyramid with base $ABCD$ and all edges of length $1$.

One has a standard 2-dimensional projection of this cube on the back face, the problem is to construct the point $P$ in this projection.

It is not that difficult to calculate that the height of $P$ above the ground plane is $\sqrt{2}/2$, so one could measure a half diagonal in the front face and use this to construct the height of $P$, but this requires the calculating of the height of $P$, which we do not want.

How can one construct the point $P$ in this 2-dimensional projection, without making calculations in advance?

• I really don't see what kind of answer you're expecting. If a monkey is drawing lines and circles randomly and he happens to construct that point, does that count ? what if his construction happens to be exactly like the one you already have ? – mercio Sep 1 '15 at 17:57
• I am hoping for some geometrical argument to show that the construction I gave (or another construction) is correct, i.e. one that does not require you to calculate the height in advance and look for that length in the cube. – Erik Rijcken Sep 1 '15 at 18:01