I want to show that in $S^2$ all self-polar triangles are congruent.
I know that if a triangle has angles $A,B,C$ and opposite sides $a,b,c$, then for its polar triangle we have :
$A'=π-a$, $Β'=π-b$, $C'=π-c$
and for its sides
$a'=π-A$, $b'=π-B$, $c'=π-C$
My intuition is that in order a triangle to be self polar it has to be
$A=B=C=π/2 $
and
$a=b=c=π/2$
(from the definition of a polar triangle)
However if we consider a triangle such that $a=π-A$ , $b=π-B$ , $c=π-C$ then this triangle has to be self-polar, since it fullfils the equalities above.
Hence, why a spherical triangle with angles $A=π/2$, $B=2π/3$, $C=π/4$
and sides $a=π/2$, $b=π/3$, $c=3π/4$
is not a self-polar triangle?
Thanks in advance for your time.