Let
be a finite set of points in three-dimensional space. Let
be the sets consisting of the orthogonal projections of the points of
onto the
-plane,
-plane,
-plane, respectively. Prove that
where
denotes the number of elements in the finite set
.
Note Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.
![\,S\,](/media/m/a/2/0/a208d81f0e71cbbb18aa0c676c46a441.png)
![\,S_{x},\,S_{y},\,S_{z}\,](/media/m/1/8/b/18bf0994bfcd475899293f4e9dbb6045.png)
![\,S\,](/media/m/a/2/0/a208d81f0e71cbbb18aa0c676c46a441.png)
![yz](/media/m/0/6/5/0659e5216349fdbdd88bf39571bb3b13.png)
![zx](/media/m/c/0/d/c0d6ff468970d42a134f2ba89d8cb5a1.png)
![xy](/media/m/5/9/6/596af52a0894be7f886bc10ba63d140f.png)
![\vert S\vert^{2}\leq \vert S_{x} \vert \cdot \vert S_{y} \vert \cdot \vert S_{z} \vert,](/media/m/8/2/e/82e75bae0c0294ecbb64856f830034a3.png)
![\vert A \vert](/media/m/8/2/7/8277d5ae1d2c1dd38bb2dbb0ac202186.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
Note Note: The orthogonal projection of a point onto a plane is the foot of the perpendicular from that point to the plane.