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The keyword you are looking for is "quantitative isoperimetric inequalities". The case of polygons was solved in the following paper: https://arxiv.org/abs/1402.4460 The "quantitative" term in the pap …

It is straightforward to see that $A',B',C'$ are reflections of the circumcenter $O$ with respect to $BC, CA,AB$. Therefore, the center of $(AOA')$ is just the intersection of the mediatrix of $OA$ wi …

Switching the roles of $ABC$ and $A'B'C'$, consider the circles passing through the vertices of a triangle $A,B,C$, midpoints of the opposite sides $A',B',C'$ and the circumcenter $O$. It is straightf …

Notice that triangles $ACD, AEB, FCB$ are similar. Working out the ratios of the sides and the angles one can see that: Triangles $AI_1I_3$ and $ACE$ are similar. Rotating $I_1I_3$ around $A$ with an …

Suppose $\omega$ is a bounded shape in $\Bbb{R}^3$ and that $\{z : (x,y,z) \in \omega \}=[0,T]$ (that is, the shape is exactly contained in the band $\{z \in [0,T]\}$. If we denote by $\omega_t = \{(x …

Here is my variant, a bit more geometrical. Denote by $P$ the original polygon, and $P_\lambda$ the contracted polygon with a factor $\lambda \in (0,1)$ which lies inside $P$. Note that $P_\lambda$ i …

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for eve …

I didn't manage to solve the problem (edit: in the meantime an answer was posted which says a precise formula using radicals cannot be found), but I can post a proof that the line joining the points w …

There is a formula for the area of the intersection of two circles of given radii in terms of the distance between the centers. The formula can be found here: http://mathworld.wolfram.com/Circle-Circl …

It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there ex …