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I know that this may be off-topic.

The gist is: I get frustrated when I solve questions with a large integral (both in physics and mathematics) or some other heavy machinery, and then see a solution which is a geometrically-inspired proof. For eg.

Let $\theta_1, \theta_2, \theta_3, … , \theta_{10}$ be positive values angles in radians such that $\theta_1+\theta_2+ \theta_3+… + \theta_{10}=2\pi$. Define complex numbers $z_1=e^{i\theta_1}$, $z_k=z_{k-1}e^{i\theta_k}$ for $2\leq k\leq 10$. Then which of these is true?
P: $|z_2-z_1|+|z_3-z_2|+…|z_1-z_{10}|\leq 2\pi.$
Q: $|z_2^2-z_1^2|+|z_3^2-z_2^2|+…|z_1^2-z_{10}^2|\leq 4\pi.$

Note that I am not asking for a solution to this question, which can be solved by considering that the perimeter of any n-sided polygon is less than the circumference of its circumscribing circle.

I want some book recommendations where these kinds of hard-but-geometrically-easy problems (algebra, complex numbers, Calculus-I,II, elementary combinatorics) are shown with beautiful geometrical interpretations.

Please keep the level to advanced high school level (should not focus too much on linear algebra (though vectors and matrices except eigenvalues are welcome), multivariable calculus, topology, graph theory, fields and rings etc.)

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  • $\begingroup$ A book that I praise is "Glimpses of algebra and geometry" by Gabor Toth, but does it correspond to your quest ? $\endgroup$
    – Jean Marie
    Jun 27 at 20:50
  • $\begingroup$ Maybe better for what you are looking for : "Thinking geometrically : a survey of geometries" by Thomas Sibley (American Mathematical Society) $\endgroup$
    – Jean Marie
    Jun 27 at 20:55
  • $\begingroup$ @JeanMarie the one you mention is a book on geometry; I wanted something where we have questions seemingly not related to geometry, but solved with unexpected links to geometry. $\endgroup$ Jun 28 at 6:08

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I haven't been able to find an exact answer, but I got some stuff which are pretty close to the required:

  1. Tristan Needham Visual Complex Analysis : See chapters "Geometry and Arthimetic", "Moebius Transformation and inversion"

  2. You can find similar problems which can be done based on Geometric intuition in exercises of Chapter-1 Needham's Visual Differential Geometry. Here is a sample.

  3. You will encounter a lot more geometry methods for algebraic problems if yo uare to learn a bit more about Conic sections. I've written quite a few answers dealing with some basic conic problems using vectors (see here)

  4. There are also another category I've encountered which are sort of like you optimize some sort of algebraic quantity through a geometric method, or, finding a geometrical problem, writing it in algebra, solving the geometrical problem independently and using that to answer the algebra problem. An example of such can be found here. The deeper idea underlying this is something known as Lagrange multipliers. You can use it to solve some more complicated problem as well like Toricelli's problem. It is discussed early on a Pavel Grinfeld's Tensor analysis book (video),

For a general answer/ a more general book , you could ask on AOPS.

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I’ll answer my own question but likely won’t accept any answer for some period of time so that I can keep getting feedback on this question.

I found this: Proofs without words, published by Mathematical Association of America

https://1lib.in/book/3418064/77fb87

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