Final answer:
The key sentence from Brown v. Board of Education declaring public school segregation unconstitutional states that 'separate educational facilities are inherently unequal,' thus violating the Fourteenth Amendment's equal protection clause.
Step-by-step explanation:
The sentence from Brown v. Board of Education that best supports the idea 'It is unconstitutional for public schools to segregate students on the basis of race' is: "We conclude that, in the field of public education, the doctrine of 'separate but equal' has no place. Separate educational facilities are inherently unequal.
Therefore, we hold that the plaintiffs and others similarly situated for whom the actions have been brought are, by reason of the segregation complained of, deprived of the equal protection of the laws guaranteed by the Fourteenth Amendment."
This statement made by the Supreme Court in 1954 clearly illustrates the view that racially segregated schools cannot provide truly equal education, thus making segregation in public education unconstitutional.