
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.

to prove
To support and demonstrate a claim or theory with facts, data, or evidence.
Der Wissenschaftler konnte seine These mit zahlreichen Studienergebnissen belegen.
The scientist was able to prove his thesis with numerous study results.

to top
To cover something, usually bread or a roll, with one or more layers of food such as cheese or sausage.
Für die Party hat sie viele Brötchen mit Lachs und Frischkäse belegt.
For the party, she topped many rolls with salmon and cream cheese.

to place
To achieve a particular place in a competition or ranking.
Die deutsche Mannschaft belegte bei der Weltmeisterschaft den zweiten Platz.
The German team placed second at the World Cup.

to impose something on someone
To impose a penalty, curse, or fee on someone. Usually used in formal or legal contexts.
Wegen des Verstoßes wurde das Unternehmen mit einer hohen Geldstrafe belegt.
Because of the violation, the company was fined heavily.
'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 express the same position in a ranking. With liegen, an expression such as „auf dem zweiten Platz“ follows; with belegen, „den zweiten Platz“ is the direct object. The statement about the rank achieved remains the same.

Unsere Mannschaft belegt nach dem letzten Spiel den zweiten Platz.
Our team took second place after the last game.
Unsere Mannschaft liegt nach dem letzten Spiel auf dem zweiten Platz.
Our team is in second place after the last game.