The incircle of a $\Delta ABC $ touches the sides $BC,CA,AB$ at $X_1,Y_1,Z_1$ ; The incircle of $\Delta X_1Y_1Z_1$touches the sides of $\Delta X_1Y_1Z_1$ at $X_2,Y_2,Z_2$ and likewise points $X_n,Y_n,Z_n$ are defined for $n > 2$.
Prove that, $∠\ Y_nX_nZ_n= 60^ \circ + (-2)^n(∠A-60^\circ) $
I tried many ways possible, but couldn't find the answer. Please help me solve this question.
Showing that $\angle X_1 = 90^\circ - \frac12 \angle A$ is just angle-chasing. You can prove this in a number of ways; for example, using the fact that $\triangle AY_1Z_1$, $\triangle BZ_1X_1$, and $\triangle CX_1Y_1$ are isosceles.
From there, the same argument applied to $\triangle X_1Y_1Z_1$ playing the role of $\triangle ABC$ shows that $\angle X_2 = 90^\circ - \frac12 \angle X_1$, and in general $\angle X_{n+1} = 90^\circ - \frac12 \angle X_n$.
If you wanted to actually find the formula for $\angle X_n$ in terms of $\angle A$, you would have to do some work. But since you're given the formula, all you have to do is verify it by induction.