What are the great examples that in interesting fashion show usefulness of Cartesian coordinates on euclidean plane?
It might be some interesting problem which has a simple solution in Cartesian coordinates or something like that, I of course appreciate all examples, but prefer examples from elementary geometry.
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I'd say "navigation/mapping". Now you might argue that the earth isn't planar, and I'd have to agree. But for a region the size of the Mediterranean, the difference from planarity is small enough that with relatively crude instruments, it's tough to measure the non-planarity, and a plane is a good-enough abstraction.
(If you move to 3D cartesian coordinates on a map, you can get to "the map that changed the world," where plotting strata beneath the UK helped to show that they were, in fact, strata, and that one could predict what one might find by digging/mining.)
To me one of the most amazing applications of geometry is the Gaussian integral: $$I = \int_{-\infty}^\infty \mathrm e^{-x^2} \mathrm d x.$$
Instead if looking at $I$, we look at $I^2$: $$I^2= \left(\int_{-\infty}^\infty \mathrm e^{-x^2} \mathrm d x\right)^2 =\int_{-\infty}^\infty \mathrm e^{-x^2} \mathrm d x\int_{-\infty}^\infty \mathrm e^{-y^2} \mathrm d y .$$ hence, $$I^2= \int_{-\infty}^\infty\int_{-\infty}^\infty \mathrm e^{-x^2 -y^2} \mathrm d x \mathrm d y .$$ Now, we see $(x,y)$ as a point in a Cartesian coordinate space; using a polar representation allows us to solve the integral easily and find that $I^2=\pi$, so $I=\sqrt{\pi}$.
To me, it's a double whammy of abstractions: intractable integral $\to$ Cartesian plane $\to$ polar representation $\to$ easy solution. Hard to beat :)