Preliminary remark: there are several possible routes around the Three Peaks. As the landscape is devoid of vegetation, the paths are easy to see. The one I suggest takes you to the Locatelli Refuge and its view of small mountain lakes. Signs indicate the direction of the refuges, making it easy to find your way.
Park in the Auronzo Refuge car park. See Practical Information for access conditions (fee applies).
(S/E) Behind the Auronzo Refuge, take a very good road suitable for vehicles. Head towards the Lavaredo and Locatelli refuges. Continue to the Lavaredo Refuge.
(1) Continue, alternating between ascents and descents, to reach a crossroads below the Locatelli Refuge. Continue up the hill to the right to reach the refuge.
(2) Return to the previous crossroads, either by taking the same path or by taking the road, which is a slight detour but allows you to enjoy the view. At the crossroads, descend to the right and then start another climb to reach the Col de Forcelina at the crossroads of the Langalm refuge.
(3) Continue slightly uphill, walk along the lakes and arrive at Col di Mezzo.
(4) Turn left to easily reach the car park of the Auronzo Refuge (S/E).

