Summary: The existence of partially positive solutions is considered for the nonlinear second-order three-point boundary value problem: \[ u''(t)+h(t)f(t, u(t))=0, \;0<t<1,\;u(0)=0,\;\alpha u(\eta)=u(1), \] when \(\alpha <0\) or \(\alpha \eta >1\). In these cases, the related Green’s function is not nonnegative. The usual cone of positive functions is not applicable. By introducing the cone of partially positive functions, the problem is transformed into a Hammerstein integral equation on the cone. The existence of one and two partially positive solutions is obtained by applying the fixed point index theorem in a cone.