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I am a sophomore in college looking forward to taking his first differential geometry course. However, I can't say that I've been an avid fan of geometry while I was in high school/middle school mostly due to bad teaching, so I am not really good when it comes to classical Euclidean geometry (I think that it is called synthetic geometry, I hope this is not a term only in my native language). By this I mean that even though I know its basic results, I can't say that I am any good at solving difficult problems that involve making some intricate constructions (to get some sense of what I mean by difficult problems in these context, take the ones given at high school level math competitions such as the ones at the USAMO) or that are slightly non-trivial (or that involve some non-standard results).

Anyway, I really liked the geometry I learned in my freshman year (mostly analytic stuff about things such as isometries and the like, some projective geometry towards the end) and I did really well. My lack of knowledge in somewhat more elementary geometry hasn't hindered me thus far, but I can't help but wonder whether for these more advanced geometry courses that I am going to take starting from this school year it would be better if I were more familiar with classical Euclidean geometry and if I should spend some more time learning that or I should just delve right into my courses.

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    $\begingroup$ While not entirely irrelevant, classical Euclidean geometry, USAMO-style is, to my knowledge, very niche for algebraic or differential geometry. $\endgroup$
    – Aphelli
    Commented Sep 8, 2021 at 11:18
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    $\begingroup$ delve right into your courses $\endgroup$
    – Арсений Кряжев
    Commented Sep 8, 2021 at 11:50
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    $\begingroup$ I had thought that the purpose of teaching Euclidean geometry in high school is to introduce students to the idea of a mathematical proof. You're already well past that point. $\endgroup$
    – David K
    Commented Sep 8, 2021 at 12:38
  • $\begingroup$ @Mindlack my English is not so great nowadays, do you want to say that those USAMO-style problems can be solved using techniques from algebraic or differential geometry? $\endgroup$
    – TheZone
    Commented Sep 8, 2021 at 18:09
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    $\begingroup$ No! USAMO-style geometry problems are usually solved through ingenious applications of elementary techniques, or some more specialized theorems that are of little other use in today’s mathematics. They’re hardly ever made easier by considering more advanced concepts as will be introduced in algebraic/differential geometry. I meant that there may be, at some points, ideas or constructions from classical Euclidean geometry that are useful for algebraic/differential geometry, but that it is not very important. $\endgroup$
    – Aphelli
    Commented Sep 8, 2021 at 18:58

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That is an interesting question. My major is quite far from Geometries, but I love to think geometrically.

My subjective opinion/answer for your question is: NO! They (Euclidean Geometry vs Algebraic Geometry, Differential Geometry) are not so related, both background and the way of thinking. In fact, AG and DG is at higher and broader level of knowledges than EG.

Nowadays, EG is almost a game of thinking; almost no deep real-life applications; almost no open and meaningful problems.

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