If the friends meet in Bergerac, the total distance travelled will
be $$87+79+61+58 \; \text{km} = 285\; \text{km}$$

If the friends meet in Bordeux, the total distance travelled will be $$87+47+31+84 \; \text{km} = 249\; \text{km}$$

If the friends meet in Langon, the total distance travelled will be $$79+47+54+37 \; \text{km}=217\; \text{km}$$

If the friends meet in Libourne, the total distance travelled will be $$61+31+54+65 \; \text{km}=211\; \text{km}$$

If the friends meet in Marmande, the total distance travelled will be$$58+84+37+65 \; \text{km}=244\; \text{km}$$

So the friends should meet in Libourne.

If the friends meet in Bordeux, the total distance travelled will be $$87+47+31+84 \; \text{km} = 249\; \text{km}$$

If the friends meet in Langon, the total distance travelled will be $$79+47+54+37 \; \text{km}=217\; \text{km}$$

If the friends meet in Libourne, the total distance travelled will be $$61+31+54+65 \; \text{km}=211\; \text{km}$$

If the friends meet in Marmande, the total distance travelled will be$$58+84+37+65 \; \text{km}=244\; \text{km}$$

So the friends should meet in Libourne.

*This problem is taken from the UKMT Mathematical Challenges.*