Question 113
An isosceles triangle $ABC$ with sides $AB = 4$, $BC = AC = 6$ is given.
The points $X_1, X_2, X_3, \dots$ in that order are taken on side $AB$, such that the lengths of segments $AX_1, X_1X_2, X_2X_3, \dots$ form an infinite geometric series with first term $3$ and common ratio $1/4$.
The points $Y_1, Y_2, Y_3, \dots$ in that order are taken on side $BC$, such that the lengths of segments $CY_1, Y_1Y_2, Y_2Y_3, \dots$ form an infinite geometric series with first term $3$ and common ratio $1/2$.
The points $Z_1, Z_2, Z_3, \dots$ in that order are taken on side $AC$, such that the lengths of segments $AZ_1, Z_1Z_2, Z_2Z_3, \dots$ form an infinite geometric series with the first term $3$ and common ratio $1/2$.
Find all triples $(a, b, c)$ of natural numbers for which the line segments $AY_a$, $BZ_b$ and $CX_c$ are concurrent.
The points $X_1, X_2, X_3, \dots$ in that order are taken on side $AB$, such that the lengths of segments $AX_1, X_1X_2, X_2X_3, \dots$ form an infinite geometric series with first term $3$ and common ratio $1/4$.
The points $Y_1, Y_2, Y_3, \dots$ in that order are taken on side $BC$, such that the lengths of segments $CY_1, Y_1Y_2, Y_2Y_3, \dots$ form an infinite geometric series with first term $3$ and common ratio $1/2$.
The points $Z_1, Z_2, Z_3, \dots$ in that order are taken on side $AC$, such that the lengths of segments $AZ_1, Z_1Z_2, Z_2Z_3, \dots$ form an infinite geometric series with the first term $3$ and common ratio $1/2$.
Find all triples $(a, b, c)$ of natural numbers for which the line segments $AY_a$, $BZ_b$ and $CX_c$ are concurrent.