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Doer vs. machucar vs. magoar

While doer, machucar, and magoar can all be translated as to hurt in English, they describe different types and sources of pain in Portuguese. Doer refers to the sensation of pain, machucar to the action of causing an injury, and magoar almost exclusively to hurting someone's feelings.

Doer

A2
Doer describes the sensation of pain or an ache. It is an intransitive verb, meaning the part of the body that hurts is the subject of the sentence. Think of it as the state of aching or being in pain.
Minha cabeça dói muito hoje.
(My head hurts a lot today.)
Depois da corrida, minhas pernas estão doendo.
(After the run, my legs are aching.)
Onde está doendo exatamente?
(Where does it hurt exactly?)
Dói de pensar na situação.
(It hurts just to think about the situation.)
Meu estômago dói quando como muito rápido.
(My stomach hurts when I eat too fast.)

Machucar

A2
Machucar refers to the action of causing a physical injury, wound, or bruise. It implies an external cause. It can be used transitively (someone hurts another) or reflexively (someone hurts themself). It can also describe a direct and sharp emotional wound.
Eu me machuquei jogando futebol ontem.
(I hurt myself playing soccer yesterday.)
Cuidado para não machucar o dedo na porta.
(Be careful not to hurt your finger in the door.)
O vidro quebrado machucou o dele.
(The broken glass injured his foot.)
Suas palavras duras me machucaram de verdade.
(Your harsh words truly hurt me.)
Este sapato novo está machucando meu calcanhar.
(This new shoe is hurting my heel.)

Magoar

B1
Magoar is used almost exclusively for emotional pain. It means to hurt someone's feelings, to offend, or to cause sadness and disappointment. It is about emotional or psychological impact, not physical injury.
Não quis te magoar com a minha piada.
(I didn't mean to hurt your feelings with my joke.)
Ela ficou magoada porque você esqueceu o aniversário dela.
(She was hurt because you forgot her birthday.)
Ele se magoa com muita facilidade.
(He gets his feelings hurt very easily.)
É terrível magoar um amigo de verdade.
(It's terrible to hurt a true friend.)
A traição dele a magoou profundamente.
(His betrayal hurt her deeply.)

Summary

In short: use doer for the feeling of an ache or an internal pain (My back hurts). Use machucar for the action of causing a physical injury or a sharp wound (I hurt my hand). Use magoar specifically for hurting someone's feelings emotionally (Your comment hurt me).