# Proof: At most 3 circles of radius 1/2 fit into the interior of a halfcircle of radius 1

It is a well known fact that at most 7 interior disjoint circles of radius 1/2 can be centered in a circle of radius 1; note that they don't need to be fully contained in the radius 1 circle.

I am interested in a simple proof idea to show that at most 3 interior disjoint circles of radius 1/2 can be centered in the interior of a halfcircle of radius 1.

While intuitively clear, I am lacking an idea as how to approach the proof and would appreciate any pointers.

• If you're assuming the first truth, and if four could fit into a half circle, how much would fit in a whole circle? – David Peterson Sep 26 '14 at 22:37
• @David Peterson, Thank you for your comment. I was thinking of a proof only involving "fundamental" geometrical facts, i.e., a self-contained proof – John P Sep 26 '14 at 22:39
• Perhaps I am misunderstanding "interior" but to me it is not a well known fact that $6$ interior disjoint circles of radius $1/2$ can be centered in a circle of radius $1$. See en.wikipedia.org/wiki/Circle_packing_in_a_circle – Henry Sep 26 '14 at 22:45
• @Henry Sorry for the confusion. 6 interior disjoint circles of radius 1/2 can not be centered in a circle of radius 1, but can if not restricted to the interior. I am on the otherhand indeed interested in the restriction of halfcircle and interior. The 6 circle part was just to provide some context. – John P Sep 26 '14 at 22:47
• What makes a circle "interior"? Does the circle have to be entirely in the bigger circle, or just its centre with part of the smaller circle being outside the bigger circle? – Henry Sep 26 '14 at 22:49