If I find something I lost, should I use 'hallar'?
For example, I found missing keys and said to my wife:
"¡Hallé las llaves!"
"¿Las encontraste?" —she replied—.
Then a discussion about 'hallar' vs 'encontrar' began.
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My knowledge of Spanish is old school so I have found that the meaning of some things have changed. "Hallar" and "Encontrar" are basically synonyms but in the most proper Mexican Spanish. "Hallar" implies knowing where something is, whereas "Encontrar" means you have what you where looking for. Exp:
"¡Hallé mis llaves! Se las llevó María." (I found my keys, Mary took them.)
"¡Encontré mis llaves! Estaban en el sofá." (I found my keys, they were in the sofa.)
"Halla" and Hallar" have the same root so you say:
"Hallé una nube que parece un pato." (I found a cloud that looks like a duck.)
"Encontré" can be used, but properly because the clouds are at a distance, "Hallar" is more fitting.
In Argentine Spanish, hallar and encontrar are exact synonyms, although hallar is not used often, just formally, as in news or papers.
Hallazgo is used though, meaning finding (as noun)
Also, both words have alternative meanings when used in reflexive*:
no me hallo meaning I don't fit, or I don't feel comfortable in a certain situation.
me encuentro aquí meaning I am here, but it's kind of formal.
* It's pretty common in Spanish that reflexive verbs have an alternative meaning than the original verb. E.G. ver = to see, verse = to look (as in you look good)
From my experience, I'd say that hallar means to find (as result of searching), while encontrar has a wider meaning, it's more like to find including finding by chance or to encounter, as in the meaning to meet someone. However, random finding could be still described as hallazgo.
At least in my country, Spain, they mean exactly the same. Even one of the definitions of "encontrar" in the official spanish dictionary has the word "hallar" in it.
Even tough, the main difference relies on the fact that you can found "hallar" in formal contexts, like scientific papers, and "encontrar" is colloquial.