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How I can add points to curve that has been plotted as parametric plot?

I am working on the brachistochrone example and I want to plot part of the cycloid, direct line and some combinations of them into same image.

How I can do this?

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closed as off-topic by Bob Hanlon, MarcoB, bbgodfrey, Young, Edmund Aug 22 '16 at 9:55

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    $\begingroup$ Have you seen the options Epilog and Prolog? You can also use Show[] to combine your plot with other Graphics[]. $\endgroup$ – Michael E2 Aug 21 '16 at 19:40
  • $\begingroup$ Sorry no I haven't. I am very new with Mathematica. I will look into these. $\endgroup$ – Zzz Aug 21 '16 at 19:45