We report on a study of three-body charmless decays B+→K+h+h− based on a 29.1 fb−1 data sample collected with the Belle detector. With no assumptions on the intermediate mechanisms, the following three-body branching fractions have been measured for the first time: B(B+→K+π−π+)=(55.6±5.8±7.7)×10−6 and B(B+→K+K−K+)=(35.3±3.7±4.5)×10−6. We present the first observation of the decay B+→f0(980)K+ with a branching fraction product of B(B+→f0(980)K+)×B(f0(980)→π+π−)=(9.6+2.5+1.5+3.4−2.3−1.5−0.8)×10−6. This is the first reported example of a B meson decay to a scalar pseudoscalar final state. We also report the first observation of B+→K*(892)0π+ decay with a branching fraction of B(B+→K*(892)0π+)=(19.4+4.2+2.1+3.5−3.9−2.1−6.8)×10−6.