An intuition that I thought of about a year ago, but my level does not allow me to prove it. I think it needs advanced tools to prove it, or just a counterexample to refute it. Who can help, please, thank you.
The locus of the point of intersection of the perpendicular tangents of a convex curve is the convex curve
Of course, the intuition is only in this direction and does not constitute an obligation, as the cardioid is a concave curve. Despite this, tracing the perpendicular tangents to it gives a circle
It turns out that the intuition is wrong. The locus of intersection of the perpendicular tangents of the curves of the figure $x^{2n}+y^{2n}=r^{2n} ،n>1$
You can view the article: https://www.researchgate.net/figure/Construction-of-the-orthoptic-of-the-Fermat-curve-x4-y41documentclass12ptminimal_fig8_337743100