Final answer:
Brown v. Board of Education was the landmark case that declared segregation unconstitutional, overturning the 'separate but equal' doctrine established by Plessy v. Ferguson and reinforcing the Equal Protection Clause of the Fourteenth Amendment.
Step-by-step explanation:
The court case that claimed segregation violated the amendment promising "equal protection under the law" is Brown v. Board of Education of Topeka, 347 U.S. 483 (1954). This historic ruling overturned the precedent established by Plessy v. Ferguson in 1896, which endorsed de jure segregation under the "separate but equal" doctrine.
The unanimous decision in Brown v. Board found that racial segregation in public schools was unconstitutional because it violated the Equal Protection Clause of the Fourteenth Amendment, thus setting a precedent that segregation was not compatible with the principles of equality enshrined in the Constitution.
Prior to this, the Plessy v. Ferguson case upheld segregation despite the Fourteenth Amendment's intentions to protect the rights of African Americans. It was not until Brown v. Board that the Supreme Court recognized the inherent inequality of segregated educational facilities and declared that even if the facilities were deemed equal, segregation alone was a violation of equal protection and was therefore unconstitutional.