The sides of a triangle ABC, inscribed in a hyperbola $xy = c^2$, makes angles $alpha, beta, gamma $ with an asymptote.

Prove that the normals at A, B, C will meet in a point if $cot2alpha + cot2beta + cot 2gamma = 0$

My approach: Assuming $A left(ct_1,frac{c}{t_1}right), B left(ct_2,frac{c}{t_2}right) and C left(ct_3,frac{c}{t_3}right)$, we can work out $t_1t_2=-cotalpha$ and so on

Equation of a normal to rectangular hyperbola is $t^3x-ty=ct^4-c$ which is fourth degree equation which doesn’t help the cause. Using determinants for concurrency doesn’t yield any fruitful results.