20030624, 10:15  #1 
Jun 2003
2^{6} Posts 
Geometry puzzle(difficult)
Here is a triangle ABC.
Circle O1 touch AB,AC.and it's in the Angle BAC Circle O2 touch AB,BC,Circle O1 and it's in the Angle ABC Circle O3 touch AC,BC,Circle O2 and it's in the Angle ACB Circle O4 touch AB,AC,Circle O3 and it's in the Angle BAC Circle O5 touch AB,BC,Circle O4 and it's in the Angle ABC Circle O6 touch AC,BC,Circle O5 and it's in the Angle ACB Prove or disprove:Circle O6 touch Circle O1 I think it's right but I can't solve it :( 
20030624, 19:42  #2 
Aug 2002
Richland, WA
2^{2}·3·11 Posts 
Correct me if I'm wrong, but before you even consider whether O6 can touch O1, the existing arrangement you have given is not planar. If you represent the sides of the triangle and the circles as nodes in a graph and try to connect them all, you can see this.
Also consider it like this: How can both O1 and O4 touch AB and AC and both have a clear path for O1 to touch O2 and O4 to touch O3? No matter how you try to arrange it, one circle in each angle has to completely block the other from touching any other circles. 
20030624, 20:13  #3 
Aug 2002
Richland, WA
2^{2}·3·11 Posts 
Ok, I realized I hadn't consider that the circles might overlap, so the problem is not as impossible as I initially thought.
However, the concept of circles "touching" seems vague. Does any intersection of circles count as touching or must the touching circles share a tangent line at the point of intersection? 
20030625, 00:54  #4 
Jun 2003
2^{6} Posts 
Share a tangent line at the point of intersection。

20030716, 13:30  #5 
Jul 2003
2×5 Posts 
Once P1 (center of O1) is chosen, all the rest of the circles are fixed, right?

20030720, 02:13  #6 
Jul 2003
2·5 Posts 
Solved it ... sort of
[I deleted the mess about GRACE because I found a better program.]
I believe I have demonstrated it, although I wouldn't call it a proof exactly. I have a construction in Wingeom format. Unfortunately the construction wasn't able to identify that O1 and O6 intersect; but it does have a measure function, so I drew a segment between the centers (P1 and P6), intersected the segment with the circles, and measured the distance between the "intersection points". The measure remains zero  indicating that O1 and O6 are tangent  as you make changes to the triangle or the initial circle size as long as all six circles remain completely within the triangle. When one of the angles is very obtuse, it is sometimes not possible to construct one or more of the circles and stay within the triangle. (I'm not positive that this is right... would appreciate verification.) http://member.newsguy.com/~rpresser/circles.jpg Trivia about the construction: [list]Placing the first circle was my first return to geometry in many years. I solved it by using an external point to construct a point along the angle bisector of BAC, and using the constructed point as the center of O1.[/list:u][list]Once a circle is placed within the angle, constructing the next circle is the LLC case of the problem of Appolonius. I found an example of this construction here. The java applet JavaSketchpad, and the details of the construction were found in the page's source as a long text parameter to the applet. Some sweat and tears and text editing and I was able to reproduce the construction in Wingeom. (I used Wingeom because it's free ... couldn't convince wife I needed GSP).[/list:u] 
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