The Simson line

Drag P on the picture. The coral dashes drop perpendicular from P to the three sides, each side extended into a line where it is too short, and the gold points are their feet D, E and F. Bring P within a few pixels of the circumcircle and it settles on it: the three feet line up, the Simson line, and the area of the triangle they make is 0 in the corner. Drag P off the circle and the feet open into a triangle whose area is |R² − OP²|/(4R²) of the triangle's (O is the centre of the circumcircle, R its radius): inside the circle it grows as P nears the centre, outside it grows as P moves away from the centre, and it is zero only on the circle. The vertices can be dragged too, and any other triangle does the same. Choose one every 3°: the 120 Simson lines of P going once round the circle are drawn, and they weave a curve with three cusps, Steiner's deltoid; the faint gold circle is the nine-point circle. One more gold point appears, the midpoint between P and the orthocentre H (with P off the circle, the point of the circle in P's direction); it always lies on the Simson line, and on the nine-point circle.