Converging Triangle

Geometry Level 5

Let there be a triangle A1A2A3A_1A_2A_3 as shown in figure below. Lines AiAi+1A_iA_{i+1} are drawn with Ai+1A_{i+1} lying on Ai1Ai2A_{i-1}A_{i-2} such that

Ai2Ai+1Ai+1Ai1=37\hspace{5cm}\Large\frac{A_{i-2}A_{i+1}}{A_{i+1}A_{i-1}} = \frac{3}{7}  i3  iN\hspace{1cm}\;i\geq 3\; i\in N

In this way Ai2Ai1Ai\triangle A_{i-2}A_{i-1}A_i becomes smaller as ii is getting larger and ultimately converges to a point say JJ. Join A1A_1 and JJ and extend it to meet A2A3A_2A_3 at EE.

\hspace{8cm} If A1JJE=ab\Large\frac{A_1J}{JE}=\frac{a}{b} where aa and bb are coprime. Find a+ba+b.


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