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Demonstration

This demonstration uses the rolling-wheel definition of cycloid, as well as the instantaneous velocity vector of a moving point, tangent to its trajectory. In the adjacent picture, P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially, P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} traverse two cycloid curves. Considering the red line connecting P 1 {\displaystyle P_{1}} and P 2 {\displaystyle P_{2}} at a given time, one proves the line is always tangent to the lower arc at P 2 {\displaystyle P_{2}} and orthogonal to the upper arc at P 1 {\displaystyle P_{1}}. Let Q {\displaystyle Q} be the point in common between the upper and lower circles at the given time. Then: