We give two new proofs of Csaki's formula for the law of the ratio 1-Q of the maximum relative to the amplitude (i.e. the maximum minus minimum) for a standard Brownian bridge. The second of these proofs is based on an absolute continuity relation between the law of the Brownian bridge restricted to the event (Q < v) and the law of a process obtained by a Brownian scaling operation after back-to back joining of two independent three-dimensional Bessel processes, each started at v and run until it first hits 1. Variants of this construction and some properties of the joint law of Q and the amplitude are described.