Incircles of triangles proving a point it passes through

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Discussion Overview

The discussion revolves around a geometry problem involving triangle ABC, a point X on segment AB, and the properties of incircles within the triangles ACX and BCX. Participants are exploring how to prove that the incircle of triangle ABC touches side AB at point X.

Discussion Character

  • Homework-related
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant outlines the problem and expresses uncertainty about how to begin the proof, mentioning the use of angle bisectors and incircles.
  • Another participant suggests that the initial diagram may be misleading and recommends redrawing it to ensure the circles touch correctly.
  • A different participant proposes adding a third small circle that touches the first two circles and the remaining sides of the triangle to create a symmetric situation.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the initial diagram's accuracy or the best approach to start the proof. Multiple viewpoints on how to visualize the problem and proceed with the proof remain present.

Contextual Notes

The discussion highlights potential limitations in the initial diagram and the need for clarity in the geometric configuration to facilitate the proof.

mxl117
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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
 

Attachments

  • 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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