The nine-point circle

Drag the three vertices. The midpoints of the sides (gold), the feet of the altitudes (blue) and the midpoints of the segments from the orthocentre H to the vertices (coral), nine points in all, lie on the gold nine-point circle; in the corner, even the farthest of them is 0.000000 off it. Its centre N stays halfway between the circumcentre O and H, its radius half the circumradius R, and the faint gold Euler line runs through O, the centroid G, N and H. Make one angle obtuse and H moves outside the triangle; all of this still holds. Make it a right angle and H lands on that vertex, two of the feet merge there too, and the circle passes through it. Choose the incircle and the three excircles: the incircle (blue, centre I, radius r) touches the three sides, and each excircle touches the lines through all three sides from outside the triangle. The nine-point circle touches all four at the coral points, the incircle from inside and the excircles from outside: NI − (R/2 − r) in the corner stays at 0, and so does the bottom row, for the excircles. The closer the triangle is to equilateral, the closer the incircle comes to the nine-point circle, and when it is exactly equilateral the two are one. Karl Wilhelm Feuerbach proved this in 1822.