位似变换与费尔巴赫定理

Homothetic Transformation and the Theorem of Feuerbach

费尔巴赫（Ｋ．Ｆｅｕｅｒｂａｃｈ）定理［１］（１０４－１１３页）是非常优美的，但它的证明是很困难的。本文给出一个简单的证法，先用位似变换的方法证明九点圆的圆心在外心和垂心连线段的中点，九点圆的半径等于外接圆半径的一半。然后通过计算九点圆与内切圆、九点圆与旁切圆的圆心距，使定理获得证明。

The theorem of Feuerbach[1]is exquisite but it is rather difficult and complex to prove.This paper provides an easy way to prove it. Firstly , the following proofs are made by means of homothet-ic transformation : 1) the centre of a nine-point circle is at the midpoint of ligature of its curcumcentre andorthocentre; 2) the radius of nine-point circle is equal to half of the radius of its curcumcircle. Then thetheorem can be proved by calculating the distances of centres between a nine-point circle and its inseribedcircle and that between a nine-point circle and its escribed circle.