情趣生活

情趣生活 => 灵机一动 => Topic started by: froid on 七月 09, 2004, 07:32:07 am

Title: 用圆覆盖三角形
Post by: froid on 七月 09, 2004, 07:32:07 am
出道比较古老的题目:

假设有个三角形可以被25个直径为2的圆完全覆盖住,证明它同样可以被100个直径为1的圆完全覆盖住。这里所说的三角形和圆都是包括边缘的实心平面图形。
Title: 用圆覆盖三角形
Post by: MrPuzzle on 七月 09, 2004, 01:18:08 pm
This is a good problem. At first, is seems to be hard, since we can't cover a circle of radius 2 with 4 circles with radius 1.  And we are lost if we go this approach. However, there's a reason that the triangle is involved. Not just the circles.

Every triangle can be partitioned into 4 equal triangles (with the same shape as the big one) by connecting middle points of each side.  Now, each of these 4 small triangles can be covered by 25 circles of radius 1 (since everything is half), thus, the big triangle can be covered by 100 small circles of radius 1.

Good problem.