Let ABC be a triangle and M a point in the plane. The perpendicular projections of M onto the three sides of the triangle are collinear if and only if M lies on the circumcircle of triangle ABC. When this is the case, the line through the three projections is called the Simson line of M.
An even more delightful property is Steiner's theorem: as M moves around the circumcircle, its Simson line envelopes a hypocycloid with three cusps. This is the curve traced by a point on a circle of radius R rolling without slipping inside a circle of radius 3R.
