A Fair Coin from a Biased One

The problem
a biased coin · P(H) = p, unknownHP(H) = pTP(T) = 1 − psimulate a perfect 50/50 outcome — how?

You have a single coin that lands heads with some unknown probability p∈(0,1)p \in (0, 1). You need to generate one fair coin flip — a single bit that is exactly 50/5050/50 — using only this biased coin.

Can you do it exactly, with no knowledge of pp?