Why are the transpose and inverse of an orthogonal matrix equal?

My answer to a question in Quora: Why are the transpose and inverse of an orthogonal matrix equal?

Orthogonal matrices have several equivalent definitions — this is reflected in previous answers to your question. It could happen that in the book that you are using, or in the lectures that you are taking, the equality of the transpose and inverse, \(A^T = A^{-1}\), is chosen as the first definition of an orthogonal matrix and its equivalence to other statements is proven later.

So my answer: depending on the chosen way of exposition of linear algebra in a book or a lecture, it could be just a definition.