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No, the midpoints of the triangle's sides would be in the same locations as the feet of the altitudes, while the Euler points (midway between the orthocenter and the reference triangle's verticies) would be distinct from them. As a result, the nine points would become only 6 distinct points.

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Q: Would the nine points for feuerbach's nine point circle all be different points if an equilateral triangle were used?
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