MPOTD - Math Problem Of The Day
-
Xiangyang Liu wrote: Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? Yes. Assume the given triangle has vertices A, B, and C. Now define points D, E, and F as the midpoints between each of the vertices. These six points can be used as the vertices of 4 smaller triangles (ADF, BDE, CFE, and DEF) that are congruent to and 1/4 the area of the original triangle ABC. These 4 triangles represent a partition of the original triangle. The idea is to cover each of the smaller triangles with 25 circles of diameter 1, and this is possible because this is just applying the known covering at a 1/2 length scale. Thus, we have covered the entire triangle with 100 circles of diameter 1.
You are right, Ray. Your proof is even mathematically complete. :)[
My articles and software tools