Hi Mehrnoosh, "to gain" means to obtain or secure something that you wanted "we gained the necessary points to win". It is often used when talking about weight "he gained weight to be heavier when playing rugby". When talking about a destination it can sometimes mean "to reach" but this doesn't happen very often. "to achieve" means to successfully bring about or reach a goal using skill and effort. "She achieved the highest grade in the class". "to reach" can mean a few things depending on the circumstances. It can mean "to stretch out" an arm in a specified direction. "she reached out her arm to help him get up". Equally it can mean "to arrive". "He finally reached the train station but the train had already left". In this sense, it can be metaphorical "she reached her goal" which is why you might have been confused about the definition. "to attain" means to succeed in meeting something. "You can attain your objectives if you are clear on what they are". Sometimes the definitions can overlap which is why it can be confusing. If you would like some more examples or you have any questions, feel free to send me a message. I hope it helped, Maddie Professional Native English Teacher

I think Maddie covered it very well. They CAN all mean the same thing, but some of them especially 'gain' and 'reach' have other meanings as well.