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Problem 91

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Right Triangles with Integer Coordinates

The points $P(x_1,y_1)$ and $Q(x_2,y_2)$ are plotted at integer co-ordinates and are joined to the origin, $O(0,0)$, to form $\mathrm{△}OPQ$.

There are exactly fourteen triangles containing a right angle that can be formed when each co-ordinate lies between $0$ and $2$ inclusive; that is, $0\le x_1,y_1,x_2,y_2\le 2$.

Given that $0\le x_1,y_1,x_2,y_2\le 50$, how many right triangles can be formed?

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格点直角三角形

考虑格点$P(x_1,y_1)$和$Q(x_2,y_2)$与原点$O(0,0)$相连构成的三角形$\mathrm{△}OPQ$。

当点$P$和点$Q$的坐标在$0$到$2$之间，也即$0\le x_1,y_1,x_2,y_2\le 2$时，恰好能构造出$14$个直角三角形。

若$0\le x_1,y_1,x_2,y_2\le 50$，能构造出多少个直角三角形？

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