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In addition, we show that if $x\\in\\mathfrak{A}$, then $v(x) = \\frac{1}{2}\\|x\\|$ if and only if $\\|x\\| = \\|\\mbox{Re}(e^{i\\theta}x)\\| + \\|\\mbox{Im}(e^{i\\theta}x)\\|$ for all $\\theta \\in \\mathbb{R}$. Among other things, we introduce a new type of parallelism in $C^*$-algebras based on numerical radius. 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First, we prove several numerical radius inequalities in $\\mathfrak{A}$. Particularly, we present a refinement of the triangle inequality for the numerical radius in $C^*$-algebras. In addition, we show that if $x\\in\\mathfrak{A}$, then $v(x) = \\frac{1}{2}\\|x\\|$ if and only if $\\|x\\| = \\|\\mbox{Re}(e^{i\\theta}x)\\| + \\|\\mbox{Im}(e^{i\\theta}x)\\|$ for all $\\theta \\in \\mathbb{R}$. Among other things, we introduce a new type of parallelism in $C^*$-algebras based on numerical radius. 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