In the figure bisectors of `/_B` and `/_C` of `DeltaABC` intersect each other in point X. Line AX intersects side BC in pont Y. `AB=5, AC=4, BC=6` then find `(AX)/(XY)`

Explanation

In `DeltaABY`,
ray BX bisects `/_ABY` ………..(Given)
`:.` by the theorem of angle bisector of a triangle,
`(AB)/(BY)=(AX)/(XY)`……………1
In `DeltaACY`
ray CX bisects `/_ACY`…..(Given)
`:.` by the theorem of angle bisector of a triangle,
`(AC)/(CY)=(AX)/(XY)` ..............2
`(AB)/(BY)=(AC)/(CY)=(AX)/(XY)`..........[From 1 and 2]
`:.5/(BY)=4/(CY)=(AX)/(XY)`
By theorem on equal ratios, ltbr `(5+4)/(BY+CY)=(AX)/(XY)`
`:.9/(BC)=(AX)/(XY)`..........(`B-Y-C`)
`:.9/6=(AX)/(XY)`
`:.(AX)/(XY)=3/2`
`(AX)/(XY)=3/2`