Let $L$ be a line in $\mathbb{C}$ that makes an angle $\alpha$ with the real axis. Let $z=x+iy$ be any point on the line $L$. Let $d$ be the shortest distance from $L$ to origin. Prove geometrically that $d=|\text{Im}(ze^{-i\alpha})|$.
My solution uses both geometry and trigonometry but I can not come with a purely geometrical solution. Is using both geometry and trigonometry in my solution allowed or I only have to use geometry? Nonetheless I would be interested to see a pure geometrical solution.
Best Answer
What do you mean by "pure" geometry? For one thing trigonometry ultimately has its root in geometry (ratio of parts of a triangle whose sides are radii of a circle).
That said, to solve this problem you just need a geometric understanding of what the map $f_{\theta}(z) = e^{i\theta}z$ does to points on the complex plane. Maybe you need Euler's formula and some trigonometry to gain that understanding -- there are perhaps other ways of going about it (i.e. remarking that $f_{\theta}$ is the exponential map on the Lie algebra of 2D rotations) but I don't see anything "impure" about the obvious approach.