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 X_{1}, Y_{1} be the points on the x-axis with the same x-coordinates as X and Y respectively, and let X_{2}, Y_{2} be the points on the y-axis with the same y-coordinates. Show that the area of the region XYY_{1}X_{1} plus the area of the region XYY_{2}X_{2} depends only on the length of the arc XY, and not on its position.

**Solution**

*Easy.*

Let O the origin. Area XYY_{1}X_{1} = area sector OXY + area OYY_{1} - area OXX_{1}, and area XYY_{2}X_{2} = area sector OXY + area OXX_{2} - area OYY_{2}. But area OXX_{1} = area OXX_{2} and area OYY_{1} = area OYY_{2}. Hence area XYY_{1}X_{1} + area XYY_{2}X_{2} = 2 area sector OXY, which is independent of the position of the arc.

© John Scholes

jscholes@kalva.demon.co.uk

12 Dec 1998