You can use PickedTriangle to get the triangle, and then calculate the distance to each of the vertices used by the triangle.
Something like this:
ent = PickedEntity()
surf = PickedSurface()
tri = PickedTriangle()
v0 = TriangleVertex(surf,tri,0)
Tformpoint vertexx(surf,v0),vertexy(surf,v0),vertexz(surf,v0),ent,0
x# = tformedx()-pickedx()
y# = tformedy()-pickedy()
z# = tformedz()-pickedz()
mind# = Sqr(x*x + y*y + z*z)
vert = v0
v1 = TriangleVertex(surf,tri,1)
Tformpoint vertexx(surf,v1),vertexy(surf,v1),vertexz(surf,v1),ent,0
x# = tformedx()-pickedx()
y# = tformedy()-pickedy()
z# = tformedz()-pickedz()
d# = Sqr(x*x + y*y + z*z)
if d<mind
mind = d
vert = v1
end if
v2 = TriangleVertex(surf,tri,2)
Tformpoint vertexx(surf,v2),vertexy(surf,v2),vertexz(surf,v2),ent,0
x# = tformedx()-pickedx()
y# = tformedy()-pickedy()
z# = tformedz()-pickedz()
d# = Sqr(x*x + y*y + z*z)
if d<mind
mind = d
vert = v2
end if
print "nearest vertex: "+vert
print "distance: "+mind
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