Well known results on near-minimax approximation using Chebyshev polynomials of the first kind are here extended to Chebyshev polynomials of the second, third, and fourth kinds. Specifically, polynomial approximations of degreen weighted by (1−x2)1/2, (1+x)1/2 or (1−x)1/2 are obtained as partial sums of weighted expansions in Chebyshev polynomials of the second, third, or fourth kinds, respectively, to a functionf continuous on [−1, 1] and constrained to vanish, respectively, at ±1, −1 or +1. In each case a formula for the norm of the resulting projection is determined and shown to be asymptotic to 4π−2logn +A +o(1), and this provides in each case and explicit bound on the relative closeness of a minimax approximation. The constantA that occurs for the second kind polynomial is markedly smaller, by about 0.27, than that for the third and fourth kind, while the latterA is identical to that for the first kind, where the projection norm is the classical Lebesgue constant λ n . The results on the third and fourth kind polynomials are shown to relate very closely to previous work of P.V. Galkin and of L. Brutman.