Let C be the circle center (0, 0), radius 1. Let X, Y be two points on C with positive x and y coordinates. Let X1, Y1 be the points on the x-axis with the same x-coordinates as X and Y respectively, and let X2, Y2 be the points on the y-axis with the same y-coordinates. Show that the area of the region XYY1X1 plus the area of the region XYY2X2 depends only on the length of the arc XY, and not on its position.
Let O the origin. Area XYY1X1 = area sector OXY + area OYY1 - area OXX1, and area XYY2X2 = area sector OXY + area OXX2 - area OYY2. But area OXX1 = area OXX2 and area OYY1 = area OYY2. Hence area XYY1X1 + area XYY2X2 = 2 area sector OXY, which is independent of the position of the arc.
© John Scholes
12 Dec 1998