We study the dynamic relations between the direct exchange rate of fiat currencies and the rate implied by bitcoin prices. The empirical results show the deviation between actual and implied rates of EUR/USD, GBP/USD, and JPY/USD affects the movements of actual and BTC-implied rates. We observe that implied rates contribute more to price discovery than actual rates before 2019 for EUR/USD and before 2018 for JPY/USD. Triangular arbitrage opportunities arise when VIX is high. The arbitrage opportunities are also related to the market capitalization of bitcoin and the trading of bitcoin futures on CBOE and CME.