'amphitheater-like' is formed from 'amphitheater' + the suffix '-like', where 'amphitheater' originates from Greek, specifically the word 'amphitheatron', with 'amphi-' meaning 'on both sides' and 'theatron' meaning 'a place for viewing'.

'amphitheatron' in Greek passed into Latin as 'amphitheatrum', then into Middle English/French as 'amphitheatre' (UK) or 'amphitheater' (US); the suffix '-like' comes from Old English 'lic' meaning 'body' or 'having the form of', forming a compound meaning 'resembling an amphitheater'.

Initially it referred literally to 'a place for viewing on both sides' (a double or surrounding theatre), and over time came to mean 'an oval or circular theatre for spectacles'; adding '-like' produced the modern sense 'resembling an amphitheater'.