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nine-point circle

Draw a triangle, any triangle (although it may be best to start with an acute triangle). Mark the midpoints of each side. Drop an altitude from each vertex to the opposite side, and mark the points where the altitudes intersect the opposite side. (If the triangle is obtuse, an altitude will be outside the triangle, so extend the opposite side until it intersects.) Notice that the altitudes intersect at a common point. Mark the midpoint between each vertex and this common point. No matter what triangle you start with, these nine points all lie on a perfect circle! This result was known to Leonhard Euler in 1765, but was rediscovered by the German mathematician Karl Feuerbach (1800–1834) in 1822.

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