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**HallsofIvy** Set up a coordinate system so that Q and P are on the x-axis and the center of the circle is the origin. Then the circle is given by equation $\displaystyle x^2+ y^2= R^2$ where "R" is the length of a radius of the circle (which you don't give but clearly the distance you want depend on R). Take point S to have coordinates (x,0). Then T has coordinates $\displaystyle (x, \sqrt{R^2- x^2})$ and P has coordinates (R, 0). The distance from S to T is $\displaystyle \sqrt{R^2- x^2}$ and the distance from S to P is R- x. So your total distance is given by $\displaystyle R- x+ \sqrt{R^2- x^2}$.