三角形內三圓兩兩相切問題

ejsoon

设想三角形內三個圓兩兩相切（外切），同時每個圓又跟三角形的兩條邊相切。

問：
一，是否任意三角形都存在這样的三個圓？

二，如果存在，可以尺規作圖吗？

三，這樣的三個圓是否圈出的面積之和最大？

前兩問是我問的，第三問已經有數學家解答了。

ejsoon

cad作圖

KeyTo9_Fans

我好像问过类似的题，参考贴子：

https://bbs.emath.ac.cn/thread-2107-1-1.html

ejsoon

第三問，已有人證明，用貪婪割圓法，即每次都割最大的圓，這樣的三個圓面積最大。

求這個證明的外網鏈接。

ejsoon

The problem of maximizing the total area of three circles in a triangle is never solved by the Malfatti circles. Instead, the optimal solution can always be found by a greedy algorithm that finds the largest circle within the given triangle, the largest circle within the three connected subsets of the triangle outside of the first circle, and the largest circle within the five connected subsets of the triangle outside of the first two circles. Although this procedure was first formulated in 1930, its correctness was not proven until 1994.

来自英文維基百科：一九三零年已經有人提出貪婪法割出的圓面積總是最大，但直到一九九四年才得到了證明。

ejsoon

ZalgallerandLos' ( 1994 ) classified all of the different ways that a set of maximal circles can be packed within a triangle; using their classification, they proved that the greedy algorithm always finds three area-maximizing circles, and they provided a formula for determining which packing is optimal for a given triangle.

来自這裏