The transition from the “early-modern” mathematical and scientific norms of establishing conventional Euclidean geometric proofs has experienced quite mixed modes of reasoning. For instance, a careful investigation based on the continued attempts by different practitioners to resolve the geometric trisectability of a plane angle suggests serious hitches with the established algebraic angles non-trisectability proofs. These faults found the root for the difficult geometric question about having straightedge and compass proofs for either the trisectability or the non-resectability of angles. One of the evident gaps regarding the norms for establishing the Euclidean geometric proofs concerns the incompatibility between the smugly asserted algebraic-geometric proofs and the desired inherent Euclidean geometric proofs. We consider an algebraically translated proof of the geometric angle trisection scheme proposed by [1]. We assert and prove that there is a complete incompatibility between the geometric and the algebraic methods of proofs, and hence the algebraic methods should not be used as authoritative means of proving Euclidean geometric problems. The paper culminates by employing the incompatibility proofs in justifying the independence of the Euclidean geometric system.