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I'm attempting to prove what is demonstrated in this Wolfram demo

Let $T_0$ be an arbitrary triangle with vertices $A_0,B_0$and $C_0$,and let $T_1$ be the triangle formed by the intersection points of the angle bisectors of $T_0$ on its three sides .Construct $T_2,T_3...$in the same manner.

Prove that the sequence$\{T_n\}_{n\geq0}$converges (in shape) to an equilateral triangle!

This question haved been solved by this paper:On Sequences of Nested Triangles.However ,this paper uses MAPLE to perform and check the calculation! So I'm curious that can we solve this question through only (mathematics) analysis?(May be ..it is difficult) Thanks.

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