Incircles of triangles proving a point it passes through

[mxl117]

Homework Statement

I am given triangle ABC and a point X on the segment AB, one circle is inscribed inside triangle ACX and another inside BCX the two circles touch at point Y which lies on the line CX. show the inscribed circle of ABC touches AB at X

Homework Equations

I suppose you could use the fact the angle bisectors of a triangle is the incenter of the incircle and circumcircle of that triangle.

The Attempt at a Solution

So far i don't have a clue, but i have drawn the diagram. See attached.
If anyone could just get me started i'd be so grateful

 

[tiny-tim]

welcome to pf!

hi mxl117! welcome to pf!

ok, your difficulty here is that your diagram really isn't helping …

the place you've put X, those circles should not be touching, so the whole diagram is misleading

start again with a new diagram, and this time draw the incircle of ABC first (that's easy!),

so that you know where X is, and your two small circles really do touch

 

[mxl117]

Thanks :)
ahh sorry i thought i'd done everything the description asked.
Do you have any hints about how to start proving it? because I'm stumped, sorry :(

 

[tiny-tim]

hi mxl117!

(i haven't solved it, but …)

i'd certainly start by drawing in a third small circle that touches the first two small circles and the remaining pair of sides of the triangle …

you now have a neat symmetric situation, with three small circles touching each other and the three sides