Suppose there is a triangle $ABC$ of sides $a, b, c$ and circumradius
$R$, such that $R ⋅ (b + c) = a \sqrt{b ⋅ c}$
We can say that ABC is (Answer:an isosceles right triangle)
By geogebra I can identify that it would be an isosceles right triangle but I could not demonstrate algebraically
By the law of sines, $a/\sin A = 2R$. Therefore, the triangle satisfies $$ \sin A = \frac{b+c}{2\sqrt{bc}} $$ Can you see why this forces the triangle to be isoceles and right?