Incircles of triangles proving a point it passes through

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In the discussion about proving that the inscribed circle of triangle ABC touches AB at point X, participants emphasize the importance of accurately drawing the diagram to reflect the problem's conditions. The initial diagram was misleading, as the circles inscribed in triangles ACX and BCX should touch at point Y on line CX. To clarify the situation, it's suggested to first draw the incircle of triangle ABC, which helps in correctly positioning point X. Additionally, introducing a third circle that touches the first two and the remaining sides of the triangle can create a symmetric scenario that aids in the proof. The conversation highlights the need for precise visual representation in geometric proofs.
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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
 

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  • incircle problem.png
    incircle problem.png
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welcome to pf!

hi mxl117! welcome to pf! :smile:

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 :wink:
 
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 :(
 
hi mxl117! :smile:

(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 :wink:
 
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