belegen
to book
To reserve or occupy a place, room, or course.
Ich habe für das nächste Semester einen Spanischkurs belegt.
I booked a Spanish course for next semester.

to book / prove / top
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to book
To reserve or occupy a place, room, or course.
Ich habe für das nächste Semester einen Spanischkurs belegt.
I booked a Spanish course for next semester.

'Belegen' is an inseparable verb with the prefix 'be-'. The prefix always remains connected to the stem 'legen' (to lay/put). The basic idea of 'laying' something onto something else helps you understand the different meanings: you put a topping on bread, a claim on a course place, or evidence onto a statement.
Sie belegt ihre Argumente mit Zahlen.
She supports her arguments with numbers.
You can recognize the exact meaning of 'belegen' from the object. If it refers to a course or a room, it means 'to reserve/book'. If it refers to a theory or statement, it means 'to prove'. If it refers to food, it means 'to cover/top'.
Ich möchte einen Italienischkurs belegen.
I would like to take an Italian course.
In the context of education and reservations, 'belegen' is often used with 'einen Kurs' (a course), 'ein Seminar' (a seminar), or 'ein Fach' (a subject) at university. It means that you have officially registered to participate.
Alle Plätze in dem Yogakurs sind bereits belegt.
All the places in the yoga course are already booked.
When it comes to competitions or rankings, 'belegen' is always combined with a placing. Typical phrases are 'den ersten Platz belegen' (to take/place first) or 'einen der vorderen Plätze belegen' (to place among the top ranks).
Die Athletin belegte im Finale den dritten Platz.
The athlete placed third in the final.
In the formal meaning 'to punish', 'belegen' has a fixed structure: Man belegt jemanden (accusative) mit etwas (dative). The 'something' is the penalty or fee.
Der Richter belegte den Angeklagten mit einer Geldstrafe.
The judge imposed a fine on the defendant.
to lie / be located
'liegen' and 'belegen' can both talk about a position in a ranking or competition. 'liegen' describes the current standing more: a team is in third place. 'belegen' sounds a bit more official and often states the result: a team places third. The words are easy to confuse because both are used with places in an order. The grammatical difference is important: you say 'auf einem Platz liegen' (to be in a place), but 'einen Platz belegen' (to place / take a place).

Bei dem Wettbewerb belegte Sofia den dritten Platz.
In the competition, Sofia placed third.
Nach dem ersten Spiel liegt die Mannschaft auf dem zweiten Platz.
After the first game, the team is in second place.