Answer:
In no way was the idea of "separate but equal" consistent with the provisions of the Fourteenth Amendment, which basically establishes equal rights between African Americans and whites in the country, and admits the citizenship of black people.
The racist governments in the south of the country sought a mechanism to, without manifestly contravening this amendment, segregate the African American population and separate it from the white population. To do this, they sought to comply with the rights established in the Constitution, but through different services from those of whites, and of much lower quality.
Thus, with the complicity of the judicial systems, they violated the principle of equality of the Constitution, protecting themselves in an alleged legality of the "separate but equal" system.