Is the $(3,4,5)$ triangle the only rectangular triangle with this property? While solving a loosely related exercise, by luck I found out that the $(3,4,5)$ triangle has the following property:
The product of the lengths ($\sqrt{2}$ and $\sqrt{5}$) of the two shorter line segments from a corner to the center of the inscribed circle equals the length ($\sqrt{10}$) of the longest one.
Somewhat satisfied with this, for me own new found, result I now wonder if any other rectangular $(a,b,c)$ triangles have this particular property. $(a,b,c)$ does not need to be a Pythagorean triple (but it would be extra nice).
It tried some straightforward algebraic equations but failed to find answer ... Maybe finding non rectangular such triangles is easier, but ideally I ask for rectangular ones.

update
Can this property of certain pythagorean triples in relation to their inner circle be generalized for other values of $n$?
is already linked to this one but I take the freedom to explicitly mention it here in post for following reasons

*

*the question asked there is about generalizing answer given here

*the answers to both questions always left some exercises for reader

*myself I am not able (I continue to try) to do these exercises

Maybe someone can fully write out the missing gaps.

In support of the answer by @heropup , all triples where
$\space B =\dfrac{A^2-1}{2}\space$ and
$\space C-B=1\space$ can be generated by
\begin{align}
A &=&&2k+1\\
B &= 2 k^2 + &&2 k\\
C &= 2 k^2 + &&2 k + 1
\end{align}
A similar set of all-primitive where
$\space C-A=2\space$ can be generated by
$\space A=4n^2-1\qquad B=4n\qquad C=4n^2+1\qquad$
et seq
$(3,4,5)\quad (15,8,17)\quad (35,12,37)
\quad (63,16,65)\quad (99,20,101)\space\cdots$
It would be interesting to see if the latter triples have properties similar to the former.
