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

 

[DeeNos]

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I would approach this problem by first identifying the key concepts and equations related to circles and triangles. From the given information, we know that there are two circles inscribed in the triangle and they touch at a point Y on the line CX. This means that Y is the point of tangency for both circles.

Next, I would use the fact that the angle bisectors of a triangle intersect at the incenter, which is the center of the inscribed circle. In this case, the angle bisectors of triangle ACX and triangle BCX intersect at point Y, which is also the point of tangency for the two circles. Therefore, Y must be the incenter of both triangles and the center of the inscribed circles.

Now, we can use the fact that the incenter is equidistant from the sides of the triangle to show that the inscribed circle of ABC touches AB at X. Since Y is the incenter, it is equidistant from the sides of triangle ACX and BCX. This means that XY is the radius of both circles and is perpendicular to the sides of the triangle at points X and Y. Therefore, the inscribed circle of ABC must touch AB at X.

In summary, we have used the concept of angle bisectors and the properties of the incenter to show that the inscribed circle of ABC touches AB at X. This is a common application of geometry in real-world problems, where we use mathematical principles to prove a point or support a hypothesis.