This is an interesting question! Of course you can always solve it by analytic geometry, and then see that all the quantities there are constructible, and construct them. But that's pretty ugly! And I don't yet see any better way to do it.
I have another construction problem which is similarly irritating to me (where I can prove with analytic geometry that it's constructible, but I can't find a nice construction, though the person who gave me the problem implied there ought to be a nice one):
Given triangle ABC, find points X and Y on AB and AC such that BX = XY = YC.