Square pyramid with integer measurements.

Question

A right square pyramid has the following dimensions:

• Length of the base square: $2a$
• Height (distance from the vertex to the base center): $h$
• Distance from the vertex to the middle point of the base edge: $b$
• Length of the pyramid's edge: $c$

The following relations hold: $ h^2+a^2=b^2$, $a^2+b^2=c^2$. Is there a way for the two triples $(h,a,b)$ , $(a,b,c)$ to be both pythagorean? If the answer is affirmative, can a parametrized family of such pairs of triples be found?

Answer 1

For this to work, $\space 1)\space$ the hypotenuse $\space (b)\space$ of one triangle must be a "leg" of the other and $\space 2)\space$ a leg $\space (a_1)\space$ of one triangle must equal the corresponding leg $\space (a_2)\space$ of the other. Here, we let $\space a_1=a_2.$ \begin{align*} h^2+a^2=b^2 &\space ,\space a^2+b^2=c^2\\ \implies b^2-h^2&=c^2-b^2 \qquad h, a<b<c \end{align*}

The problem is in condition $2)$ where $\space a_1=a_2,\space a_1,a_2\in\mathbb{N}.\space $ It would be easy if $\space a_1 \ne a_2\space $ and we let $\space a_1\space$ be the distance from the middle of the pyramid to the base and let $\space a_2\space$ be the [different] half-length of the base such as

\begin{align*} (h,a_1,b)=(3,4,5),\space (b,a_2,c)=(5,12,13)&\implies base=6a_1\space \space \\ (h,a_1,b)=(5,12,13),\space (b,a_2,c)=(13,84,85)&\implies base=14a_1\\ (h,a_1,b)=(7,24,25),\space (b,a_2,c)=(25,312,313)&\implies base=26a_1\\ (h,a_1,b)=(9,40,41),\space (b,a_2,c)=(41,840,841)&\implies base=42a_1\\ (h,a_1,b)=(15,8,17),\space (b,a_2,c)=(17,144,145)&\implies base=36a_1\\ (h,a_1,b)=(21,20,29),\space (b,a_2,c)=(29,420,421)&\implies base=42a_1\\ \end{align*}

There are an infinite number of these where, for $A^2+B^2=C^2,\quad $ $\space C_n=A_{n+1}\space $

If $a_1 = a_2\space$ can be non-integers, we do have tuples that are not Pythagorean triples (which require all sides to be integers) but which do fit the Pythagorean Theorem, e.g.

 a^2    h   b   c 
 24     1   5   7 
 96     2   10  14 
 216    3   15  21 
 129    7   13  17 
 240    7   17  23 

but, if the "rem" is removed from the program lines below, no integer values for $\space (a)\space $ are found in testing $h,a<b<c \le 100,000$.

 110 print " Hi number";
 120 input x1
 130 print "a^2",,"h ","b ","c "
 140  for h5 = 1 to x1
 150  for b5 = h5+1 to x1
 160  for c5 = b5+1 to x1
 170     if b5^2-h5^2 = c5^2-b5^2
 180         a5 = sqr(c5^2-b5^2)
 190  rem if a5 = int(a5)
 200         print c5^2-b5^2,;
 210         if c5^2-b5^2 < 100 : print ,; : endif
 220         print h5,b5,c5
 230  rem  endif
 240    endif
 250    next c5
 260    next b5
 270    next h5