A vane arc segment comprising: an airfoil piece defining first and second airfoil platforms (64, 66) and an airfoil section extending between the first and second airfoil platforms (64, 66), the airfoil section having a trailing edge, a leading edge, a pressure side, and a suction side, the first and second airfoil platforms (64, 66) each defining a first airfoil platform circumferential side (64c, 66c), a second airfoil platform circumferential side (64c, 66d), a forward airfoil platform side (64a, 66a), and an aft airfoil platform side (64b, 66b), a gaspath side (64e, 66e), a non-gaspath side (64f, 66f), and the non-gaspath side (64f, 66f) of at least one of the first or second airfoil platforms (64, 66) having a radial flange projecting there from, the radial flange extending continuously and including a flange first leg portion extending adj acent the trailing edge, a flange second leg portion extending from the flange first leg portion and curving around the suction side, and a flange third leg portion extending from the flange second leg portion toward the forward airfoil platform side (64a, 66a); and a vane support system comprising: first and second support platforms (72a, 78) for supporting the airfoil piece there between, each of the first and second support platforms (72a, 78) defining a forward support platform side, an aft support platform side, a first support platform circumferential side, a second support platform circumferential side, an airfoil side, and a non-airfoil side opposite the airfoil side, the airfoil side of at least one of the first or second support platform (72a, 78) having a radial channel therein being complementary to the radial flange and receiving the radial flange, the radial channel including a radial channel first leg portion extending adjacent the aft support platform side from a location at or near one support platform circumferential side (84c, 84d) toward the other support platform circumferential side (84c, 84d), a radial channel second leg portion extending from an end of the radial channel first leg portion and curving toward the forward support platform side, and a radial channel third leg portion extending from an end of the second leg portion toward the forward support platform side. an airfoil piece defining first and second airfoil platforms (64, 66) and an airfoil section extending between the first and second airfoil platforms (64, 66), the airfoil section having a trailing edge, a leading edge, a pressure side, and a suction side, the first and second airfoil platforms (64, 66) each defining a first airfoil platform circumferential side (64c, 66c), a second airfoil platform circumferential side (64c, 66d), a forward airfoil platform side (64a, 66a), and an aft airfoil platform side (64b, 66b), a gaspath side (64e, 66e), a non-gaspath side (64f, 66f), and the non-gaspath side (64f, 66f) of at least one of the first or second airfoil platforms (64, 66) having a radial flange projecting there from, the radial flange extending continuously and including a flange first leg portion extending adj acent the trailing edge, a flange second leg portion extending from the flange first leg portion and curving around the suction side, and a flange third leg portion extending from the flange second leg portion toward the forward airfoil platform side (64a, 66a); and a vane support system comprising: first and second support platforms (72a, 78) for supporting the airfoil piece there between, each of the first and second support platforms (72a, 78) defining a forward support platform side, an aft support platform side, a first support platform circumferential side, a second support platform circumferential side, an airfoil side, and a non-airfoil side opposite the airfoil side, the airfoil side of at least one of the first or second support platform (72a, 78) having a radial channel therein being complementary to the radial flange and receiving the radial flange, the radial channel including a radial channel first leg portion extending adjacent the aft support platform side from a location at or near one support platform circumferential side (84c, 84d) toward the other support platform circumferential side (84c, 84d), a radial channel second leg portion extending from an end of the radial channel first leg portion and curving toward the forward support platform side, and a radial channel third leg portion extending from an end of the second leg portion toward the forward support platform side. first and second support platforms (72a, 78) for supporting the airfoil piece there between, each of the first and second support platforms (72a, 78) defining a forward support platform side, an aft support platform side, a first support platform circumferential side, a second support platform circumferential side, an airfoil side, and a non-airfoil side opposite the airfoil side, the airfoil side of at least one of the first or second support platform (72a, 78) having a radial channel therein being complementary to the radial flange and receiving the radial flange, the radial channel including a radial channel first leg portion extending adjacent the aft support platform side from a location at or near one support platform circumferential side (84c, 84d) toward the other support platform circumferential side (84c, 84d), a radial channel second leg portion extending from an end of the radial channel first leg portion and curving toward the forward support platform side, and a radial channel third leg portion extending from an end of the second leg portion toward the forward support platform side.