Hello, blacksuzaku!

The coffee shop is 655 meters from the grocery store,

393 meters from the bookshop, and 314 meters from the tavern.

The bookshop is 236 meters from the tavern and 524 meters from the grocery.

Assuming the distance between the bar and the grocery is less than

the distance between the coffee shop and the grocery,

what is the distance between the bar and the grocery,

rounded to the nearest meter?

If I interpret the question correctly,

. . the tavern $\displaystyle (T)$ is inside the triangle $\displaystyle CGB.$

Since $\displaystyle CB^2 + BG^2 \:=\:CG^2\;(393^2 + 524^2\:=\:655^2),\;\angle B = 90^o$

. . In fact, $\displaystyle \Delta CGB$ is a 3-4-5 right triangle. Code:

C
*
|**
| * *
| * *
| * *
| *314 *
393 | * *
| * * 655
| *T *
| * * *
| α *236 x * *
| * θ * *
B *-----------------------* G
524

Let $\displaystyle \alpha = \angle CBT,\;\theta = \angle TBG,\;x = TG$

Use the Law of Cosines in $\displaystyle \Delta CBT$:

. . $\displaystyle \cos\alpha \:=\:\frac{393^2 + 236^2 - 314^2}{2(393)(236)} \:=\:0.601355285$

. . Hence: .$\displaystyle \alpha \:\approx\:53^o\quad\Rightarrow\quad\theta \:\approx\:37^o$

Use the Law of Cosines in $\displaystyle \Delta TBG$:

. . $\displaystyle x^2\:=\:236^2 + 524^2 - 2(236)(524)\cos37^o\:=\:132747.0766$

Therefore: .$\displaystyle x\:\approx\:364$ m.