Find a necessary and sufficient angle condition (independent of $a,b,c$ -- see under "what I have got so far" for examples) such that $a^2+c^2=nb^2$ where $n$ is a positive integer.
Note: As usual $BC=a$, $CA=b$, $AB=c$.
What I have got so far:
$a^2+c^2=b^2$ is equivalent to $\angle ABC=90^o$.
$a^2+c^2=2b^2$ is equivalent to $\angle LAC=\angle ABM$ and $\angle ANC = \angle ALB$, where $L,M,N$ are midpoints of the sides $BC,CA,AB$. (Is there a nicer angle condition than this? The "dual" nature of it is slightly unsettling in comparison to the simple "$\angle ABC=90^o$" of the $a^2+c^2=b^2$ case, and the "medians perpendicular" of the $a^2+c^2=5b^2$ case.)
$a^2+c^2=3b^2$. ??? (The answers here show that this is equivalent to a length condition in the triangle. How does this length condition convert into a nice angle condition within the triangle, like the ones above?)
$a^2+c^2=4b^2$. ???
$a^2+c^2=5b^2$ is equivalent to the medians $CN$ and $AL$ being perpendicular, where $N$ is the midpoint of $AB$ and $L$ is the midpoint of $BC$.