Triangles on a chess board

[From the notes of Emily Moore. Modified by Sam Rebelsky.]

A chess board is an array of 8 by 8 squares. In an earlier exercise, we reflected on the number of squares one could draw on the chess board. In this exercise, we will consider the number of triangles we can draw on the chess board.

For the purposes of this problem, the chessboard is arranged so that the squares are aligned with the horizontal and vertical axes (i.e., the board is "straight").

How many unique triangles can we draw on the chessboard if each corner of every triangle must fall at the corner of a square on the chessboard?

How many unique triangles can we draw on the chessboard if each corner of every triangle must fall at the corner of a square on the chessboard, one edge of each triangle must be horizontal, and one edge of each triangle must be vertical?

How many unique triangles can we draw on the chessboard if each corner of every triangle must fall at the corner of a square on the chessboard, one edge of each triangle must be horizontal, one edge of each triangle must be vertical, and the horizontal and vertical edges of each triangle must be of equal length?


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