Three isosceles triangles $△ABC'$, $△AB'C$, and $△A'BC$ were constructed on the sides of triangle $ABC$. The vertex angles have the following measurements; $∠A'=α, ∠B'=β, ∠C'=γ$. Prove that if $α+β+γ=360°$ then the measurements of the angles of $△A'B'C'$ do not depend on the measurements of the angles of $△ABC$.
$D$, $E$, and $F$ are the midpoints of the sides of $△ABC$, making $A'E$, $B'F$, and $C'D$ the perpendicular bisectors of their respective triangles.
Let us consider the circles $\Gamma_A,\Gamma_B,\Gamma_C$ centered at $A',B',C'$ with radii $A'B,B'C,C'A$.
The constraint on the angles ensures that these circles meet at a point $P$.
We also have that $PB'AC', PC'BA'$ and $PA'CB'$ are kites, therefore $\widehat{B'A' C'}=\frac{1}{2}\alpha$ and so on.