The clue lies in the placement of the USD in each pair.
For the EURUSD, the USD is the quote currency. While for the USDCHF, the USD is the base currency.
Thus, when the EURUSD is trending down it means that the USD is strengthening. If the USD is strengthening, you should see the USDCHF trending up. This is how the EURUSD and USDCHF are negatively correlated.
Keep this in mind if you intend to place trades on these pairs. For example, if you go Long on the EURUSD, you should avoid going Long on the USDCHF as well, since they are negatively correlated. You must also avoid double exposure, e.g., going Long on the EURUSD and Short on the USDCHF at the same time, because you are essentially betting on the USD to weaken for both trades.
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