Final answer:
The tibial branch supplying blood to the lateral third toe is likely part of the lateral plantar artery, which originates from the bifurcation of the posterior tibial artery.
Step-by-step explanation:
The tibial branch referred to in the question likely pertains to the branches of the posterior tibial artery that supplies blood to the foot. The posterior tibial artery itself is a branch from the popliteal artery that arises posterior to the knee and is responsible for supplying blood to the posterior compartment of the leg and plantar surface of the foot. Specifically, the artery bifurcates into the medial and lateral plantar arteries, with the lateral plantar artery continuing further to supply the lateral three toes. The anatomical structures mentioned, such as the anterior border of the tibia, tibial tuberosity, interosseous membrane, and soleal line, are critical for muscle attachment and structural integrity but are not directly associated with the branching of the arterial supply to the toes.