begegnen
to meet
To meet someone by chance. This verb is used with the dative.
Gestern bin ich in der Stadt meiner alten Schulfreundin begegnet.
Yesterday I met my old school friend in town.

to encounter / meet
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to meet
To meet someone by chance. This verb is used with the dative.
Gestern bin ich in der Stadt meiner alten Schulfreundin begegnet.
Yesterday I met my old school friend in town.

The verb 'begegnen' (to encounter/meet) always requires the dative. You encounter 'jemandem' (someone, dative) and not 'jemanden' (someone, accusative). This is an important rule and a common source of mistakes.
Auf dem Weg zur Arbeit bin ich meinem Chef begegnet.
On the way to work, I ran into my boss.
The perfect tense of 'begegnen' is always formed with the auxiliary verb 'sein' (to be), not with 'haben' (to have). So the correct form is 'ich bin begegnet' (I have encountered/met).
Bist du ihm danach noch einmal begegnet?
Did you meet him again after that?
'Begegnen' is an inseparable verb. The prefix 'be-' remains firmly attached to the verb stem in all conjugated forms and sentence positions. The past participle is 'begegnet' (without 'ge-').
Ich hoffe, dass ich ihr morgen auf der Konferenz begegne.
I hope that I meet her tomorrow at the conference.
In the figurative meaning, 'begegnen' is often used with abstract nouns in the dative, for example 'einer Gefahr begegnen' (to confront a danger), 'einem Problem begegnen' (to address a problem), or 'einer Herausforderung begegnen' (to face a challenge).
Das Unternehmen muss den neuen Herausforderungen des Marktes begegnen.
The company must address the new challenges of the market.
To express the manner of treatment, 'begegnen' is often combined with an adverb (e.g., 'freundlich' (kindly), 'offen' (openly)) or a prepositional phrase (e.g., 'mit Respekt' (with respect)).
Sie begegnet neuen Ideen immer sehr offen.
She always approaches new ideas very openly.
to meet / to hit
'treffen' and 'begegnen' can both mean that two people come together in one place. The most important difference is: 'treffen' can be planned or by chance. It is often used for appointments or arrangements to meet. 'begegnen' usually means that you see or meet someone by chance. Also, 'begegnen' takes the dative: Man begegnet jemandem. Learners can easily confuse the words because both describe an encounter. If there is an arrangement to meet, 'treffen' sounds more natural. If it happens by chance, 'begegnen' is often a better fit.

Auf dem Weg zur Schule begegnet Anna ihrer alten Lehrerin.
On the way to school, Anna runs into her old teacher.
Nach der Arbeit treffe ich meinen Bruder vor dem Kino.
After work, I'm meeting my brother in front of the cinema.