Thanks very much, Kuba. Both the approaches you suggest work, but the latency time is too large. For all intents and purposes, working with a dynamic locator on the cloud seems to be out of the question. The following piece of code is a small part of the kind of situation I'm dealing with, where the red vertex at the top should move smoothly as you drag it around. So far, I have changed the control to a 2D slider, which improves only slightly on the locator performance.
Manipulate[Module[{pA = {-1, 0},
pB = {1, 0}},
loc = Locator[
Dynamic[pC, (pC = {Min[Max[#[[1]], -1.5], 1],
Min[Max[#[[2]], 0.3], 2]}) &],
Graphics[{Red, EdgeForm[Black], Disk[{0, 0}, 0.1]},
ImageSize -> 10]];
Graphics[{Style[Text["A", pA, {1, 1}], 12],
Style[Text["B", pB, {-1.5, 0}], 12],
Style[Text["C", pC, {0, -2}], 12], {Yellow,
Triangle[{pA, pB, pC}]},
{Thickness[0.005], Line[{pC, pA, pB, pC}]},
loc},
PlotRange -> {{-2, 2}, {-0.5, 3.0}},
ImageSize -> 500]], {{pC, {0.22, 1.1}}, ControlType -> None}]