Pyramid with unit sides inside a cube

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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?