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In this paper, we apply the variational method with the Structural PrescribedBoundary Conditions (SPBC) to prove the existence of periodic andquasi-periodic solutions for planar four-body problem with $m_1=m_3$ and$m_2=m_4$. A path $q(t)$ in $[0,T]$ satisfies SPBC if the boundaries $q(0)\in\mathbf{A}$ and $q(T)\in \mathbf{B}$, where $\mathbf{A}$ and $\mathbf{B}$ aretwo structural configuration spaces in $(\mathbf{R}^2)^4$ and they depend on arotation angle $\theta\in (0,2\pi)$ and the mass ratio $\mu=\frac{m_2}{m_1}\in\mathbf{R}^+$.  We show that there is a region $\Omega\subseteq (0,2\pi)\times R^+$ such thatthere exists at least one local minimizer of the Lagrangian action functionalon the path space satisfying SPBC $\{q(t)\in H^1([0,T],$ $(\mathbf{R}^2)^4)| $$q(0)\in $ $\mathbf{A}, q(T)\in $ $\mathbf{B}\}$ for any $(\theta,\mu)\in\Omega$. The corresponding minimizing path of the minimizer can be extended toa non-homographic periodic solution if $\theta$ is commensurable with $\pi$ ora quasi-periodic solution if $\theta$ is not commensurable with $\pi$. In thevariational method with SPBC, we only impose constraints on boundary and we donot impose any symmetry constraint on solutions. Instead, we prove that oursolutions extended from the initial minimizing pathes have the symmetries.  The periodic solutions can be further classified as simple choreographicsolutions, double choreographic solutions and non-choreographic solutions.Among the many stable simple choreographic orbits, the most extraordinary oneis the stable star pentagon choreographic solution when$(\theta,\mu)=(\frac{4\pi}{5},1)$. Remarkably the unequal-mass variants of thestable star pentagon are just as stable as the basic equal mass choreography(See figure 1).

A continuum of periodic solutions to the four-body problem with various choices of masses