A surprising fact about
triangles which wasn't known until the 19th century:
Take any triangle, and draw a line from each vertex which meets the opposite side at a right angle. (These lines are called "altitudes", and all pass through a single point in the triangle's interior which is called the "orthocentre".) Then the feet of these altitudes--i.e., the points where they meet the sides of the triangle--and the midpoints of the triangle's sides and the midpoints of the lines joining the vertices to the orthocentre (nine points in total) all lie on a single circle.