Characterizing triangles with integer-degree angles that are similar to any of their iterated orthic triangles A triangle has integer angles $a,b,c$ (in degrees) and we create the triangle's pedal triangle and call it Pedal-$(1)$. (Here, "pedal triangle" means the specific pedal triangle whose vertices are the feet of the altitudes of the original triangle; in other words, the orthic triangle.)
Pedal-$(n)$'s pedal triangle is Pedal-$(n+1)$.
If the original triangle is similar to any Pedal-$(n)$'s we call it a "x" triangle.
I got if a triangle is a "x" triangle then $a,b,c$ must be divisible by $4$.
But not that if $a,b,c$ are divisible by $4$ then the triangle must be a "x" triangle.
And trying all the trios for $a,b,c$ divisible by $4$ shows that every triangle in this category is indeed a "x" triangle?
Why does this happen? 
 A: If Pedal$(n)$ has angles $(a,b,c)$, then Pedal$(n+1)$ has angles:
$$
\cases{
(180°-2a,\quad 180°-2b,\quad 180°-2c)&if Pedal$(n)$ is an acute triangle,\\
(2a-180°,\quad2b,\quad2c)&if angle $a$ is obtuse.\\}
$$
That of course explains why all "x" triangle must have $a$, $b$, $c$ divisible by $4$. 
On the other hand, a triangle $T$, with angles $a$, $b$, $c$ divisible by $4$, is not an "x" triangle only if there is no other triangle with angles divisible by $4$ which has $T$ as orthic triangle. But that is never the case, for $T$ is the orthic triangle of a triangle having angles $a'$, $b'$, $c'$ (also divisible by $4$) given by:
$$
\cases{
a'=90°-a/2,\quad b'=90°-b/2,\quad c'=90°-c/2 & if none of $a$, $b$, $c$ is divisible by $8$,\\ 
a'=90°+a/2,\quad b'=b/2,\quad\quad\quad\ c'=c/2 & if $b$ and $c$ are divisible by $8$ while $a$ is not.\\ 
}
$$
No other case is possible, because $a+b+c=180°$ is not divisible by $8$.
In other words: in every chain of nested orthic triangles, if the angles of a triangle are divisible by $4$, then the angles of the triangles preceding and following it in the chain are given uniquely by the relations above. Such a chain cannot then have a first or last triangle, nor branches, and sooner or later a triangle with the same angles as the first one must appear.
