The Supreme Court felt that the "separate but equal" doctrine didn't violate the 14th Amendment because they believed that both blacks and whites were being treated equally despite being separate. The 14th Amendment gave American citizenship to all African Americans, this included former slaves. This means that the government must give all American citizens the same rights, equal protection, and due process.
The Brown v. Board of Education case proved that the "separate but equal" doctrine was false. This was where a young African American girl had to walk a long way to her black school, while there was a white school very close to her home. This was an obvious problem and proved that "separate but equal" was not equal at all. Eventually, this case ended segregation in public schools. The Plessy v. Ferguson case also proved this, but the Supreme Court tried to justify the "separate but equal" doctrine by saying how blacks and whites have the same equal facilities.
