1994 CMO 第 6 题

Let $M$ be a point which has coordinates $(p\times 1994,7p\times 1994)$ in the Cartesian plane ( $p$ is a prime). Find the number of right-triangles satisfying the following conditions:

(1) all vertexes of the triangle are lattice points, moreover $M$ is on the right-angled corner of the triangle;

(2) the origin ( $0,0$ ) is the incenter of the triangle.

令 $M$ 为笛卡尔平面中坐标为 $(p\times 1994,7p\times 1994)$ 的点($p$ 是素数)。找出满足以下条件的直角三角形的数量：

(1)三角形的所有顶点都是格点，且$M$在三角形的直角上；

(2) 原点 ( $0,0$ ) 是三角形的内心。