Use 'eureka' to express a successful discovery or a sudden insight after much effort. After checking the code for days, eureka! The bug was fixed.
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Historical Origin
'Eureka' has Greek roots, famously used by Archimedes when discovering water displacement. Just like Archimedes, I shouted 'eureka' when I solved the puzzle.
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Casual Usage
It's informal, suitable for casual conversation rather than formal or academic writing. I was looking for my keys and then, eureka! They were in my jacket pocket.