# Is there any pyramid like that?

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1. Oct 9, 2015

### karosz

I had geometry quite a while ago and I wonder if anyone has any idea how to tackle this problem:

Is there any ABCDS pyramid (where ABCD is a rectangle) in which each 2 edges have different lengths and |AS|+|CS|=|BS|+|DS|

Thanks

2. Oct 9, 2015

### Staff: Mentor

Welcome to PF!

Do you mean a real pyramid like the ones in Egypt or mathematically?

3. Oct 9, 2015

### karosz

mathematically. Something tells me one can construct such a pyramid, but I cant give explicit example..

4. Oct 9, 2015

### BvU

No Egyptian pyramids I know of have a rectangular floor plan....

I assume the unequal edges are only the non-horizontal ones (because of this rectangular floor plan) ?

In that case I look at a rectangle on the floor from above; then I imagine two equal length (length > diagonal of rectangle) wires, one from A to C and one from B to D. If I lift them with a hook or something, where can I end up with the top of the pyramid ? Can the projection of the tip on the floor be off both symmetry axes or only off one ?

5. Oct 9, 2015

### karosz

Yes, only non horizontal ones. I tried the same but somehow its hard to show anything "mathematically"

6. Oct 9, 2015

### Staff: Mentor

You could draw the rectangular base and attach threads to each corner and then pull them up to a common point. This is how I'd visualize it.

7. Oct 9, 2015

### karosz

Me too, I'll try it. But how it can be proven using elementary geometry?

8. Oct 9, 2015

### Staff: Mentor

I don't know how to prove it other than starting with the rectangular base and drawing a line segment of a any length perpendicular to the base such that one endpoint is in the rectangle then you can draw a line segment from any vertice of the rectangular base to the free line segment endpoint. Something like that...

9. Oct 9, 2015

### karosz

10. Oct 9, 2015

### Staff: Mentor

Are you trying to proof that such a shape exists?

11. Oct 9, 2015

### karosz

I firmly believe that it exists - I did an experiment with strings

But I cant guarantee it exists