If the day before yesterday was a Saturday, determine what day of the week it will be exactly three days after tomorrow.

Let's anchor ourselves to the current day timeline:
* Given: Day before yesterday = Saturday
* Therefore, Yesterday = $\text{Saturday}+1\text{ day}=\text{Sunday}$
* Therefore, Today = $\text{Sunday}+1\text{ day}=\text{Monday}$
* Therefore, Tomorrow = $\text{Monday}+1\text{ day}=\text{Tuesday}$

We need to find the day exactly three days after tomorrow:

$$\text{Target Day}=\text{Tomorrow}+3\text{ days}$$

$$\text{Target Day}=\text{Tuesday}+3\text{ days}=\text{Friday}$$

Hence, the day will be Friday.