2001 AMC 8 Problems/Problem 23
Points , and are vertices of an equilateral triangle, and points , and are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices?
Solution 1 (Complimentary Counting)
There are points in the figure, and of them are needed to form a triangle, so there are possible triplets of the points. However, some of these created congruent triangles, and some don't even make triangles at all.
Case 1: Triangles congruent to There is obviously only of these: itself.
Case 2: Triangles congruent to There are of these: and .
Case 3: Triangles congruent to There are of these: and .
Case 4: Triangles congruent to There are again of these: and .
However, if we add these up, we accounted for only of the possible triplets. We see that the remaining triplets don't even form triangles; they are and . Adding these into the total yields for all of the possible triplets, so we see that there are only possible non-congruent, non-degenerate triangles,
Solution 2 (Brute Force)
We can do casework in this problem like the solution above, but that would take too much time. Instead, we see that we can split this equilateral dot triangle to two halves by dropping an altitude from the top vertex of the big triangle. Using the smaller triangle we won't have to worry about extra unneeded cases. We can see that there are three distinct triangles in the half, and combining this with the larger equilateral triangle our answer is , which is
Solution 3 (elimination)
Notice that 20 is obviously too high (There are only 20 ways to choose 3 of the points to form a triangle or a line in total!) and you can count 4 distinct triangles quickly: , , , . So the answer is
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