Using `RegionNearest`
-------------------------------

Imagine the triangle as a one-dimensional region, `r1`, a line, embedded in  a plane.


    r1 = Line[{{0, 0}, {3, 1}, {2, 0}, {0, 0}}];
    RegionDimension[r1]
    RegionEmbeddingDimension[r1]

>1  
>2

Get the radius of a circle, with the triangle centroid as center, that intersects the farthest vertex of the triangle.

    c = RegionCentroid[r1];  (* the gray point *)
    radius = Max[EuclideanDistance[c, #] & /@ {{0, 0}, {3, 1}, {2, 0}}];
    
Animate a black point going around the circle.

And display a (red) point on the triangle that is currently nearest to the black point on the circle. 

    Animator[Dynamic[n], {0, N[2 Pi], .01}]
    Graphics[{r1, AbsolutePointSize[10],
      Gray, Point[c],
      Black, Dynamic@ Point[d = radius {Cos[n], Sin[n]} + c],
      {Red, Dynamic@Point[RegionNearest[r1, d]]},
      {Dashed, Circle[c, radius]}}]

![animator][1]


  [1]: https://i.sstatic.net/pmSkK.png