This study explores the phenomenon of shape coexistence in nuclei around \(^{172}\) Hg, with a focus on the isotopes \(^{170}\) Pt, \(^{172}\) Hg, and \(^{174}\) Pb, as well as the \(^{170}\) Pt to \(^{180}\) Pt isotopic chain. Utilizing a macro-microscopic approach that incorporates the Lublin–Strasbourg Drop model combined with a Yukawa-Folded potential and pairing corrections, we analyze the potential energy surfaces (PESs) to understand the impact of pairing interaction. \(\hbox {For}\,\) 170Pt, the PES exhibited a prolate ground state, with additional triaxial and oblate-shaped isomers. In \(^{172}\) Hg, the ground-state deformation transitions from triaxial to oblate with increasing pairing interaction, demonstrating its nearly \(\gamma\) -unstable nature. Three shape isomers (prolate, triaxial, and oblate) were observed, with increased pairing strength leading to the disappearance of the triaxial isomer. \(^{174}\) Pb exhibited a prolate ground state that became increasingly spherical with stronger pairing. While shape isomers were present at lower pairing strengths, robust shape coexistence was not observed. For realistic pairing interaction, the ground-state shapes transitioned from prolate in \(^{170}\) Pt to a coexistence of \(\gamma\) -unstable and oblate shapes in \(^{172}\) Hg, ultimately approaching spherical symmetry in \(^{174}\) Pb. A comparison between Exact and Bardeen–Cooper–Schrieffer (BCS) pairing demonstrated that BCS pairing tends to smooth out shape coexistence and reduce the depth of the shape isomer, leading to less pronounced deformation features. The PESs for even–even \(^{170-180}\) Pt isotopes revealed significant shape evolution. \(^{170}\) Pt showed a prolate ground state, whereas \(^{172}\) Pt exhibited both triaxial and prolate shape coexistence. In \(^{174}\) Pt, the ground state was triaxial, coexisted with a prolate minimum. For \(^{176}\) Pt, a \(\gamma\) -unstable ground state coexists with a prolate minimum. By \(^{178}\) Pt and \(^{180}\) Pt, a dominant prolate minimum emerged. These results highlight the role of shape coexistence and \(\gamma\) -instability in the evolution of nuclear structure, especially in the mid-shell region. These findings highlight the importance of pairing interactions in nuclear deformation and shape coexistence, providing insights into the structural evolution of mid-shell nuclei.
