Circle, inscribed in $ABC$, touches $BC, CA, AB$ in points $A’, B’, C’$. $AA’ BB’, CC’$ intersect at $G$. Circumcircle of $GA’B’$ crosses the second time lines $AC$ and $BC$ at $C_A$ and $C_B$. Points $ A_B, A_C, B_C,B_A, C_A, C_B$ are concyclic. Looks straightforward, but I’m struggling to get something out of the given. […]

We have $n$ points on the unit circle. What is the best configuration if we want to maximize the sum of the pairwise distances of the given points?

Suppose that an ellipse is rolling along a line. If we follow the path of one of the foci of the ellipse as it rolls, then this path formes a curve – namely an undulary. Now consider the following diagram The points $F$ and $F’$ are the forci of the ellipse, which is rolling along […]

Let $W$ be a set of points in $\mathbb{R}^n$. Let $C$ be the convex hull of the members of $W$. Is there a simple way of demonstrating that for any $x \in C$ and any $y \in \mathbb{R}^n \backslash C$, there must be some $w \in W$ such that $\|w – x\| < \|w – […]

I’m tutoring high school students in Math for a local College and Career prep program and would like to have a reference book on hand that I can consult. I’m a Comp Sci graduate so I have a pretty strong background in Math but it’s been a while since I used high school level Algebra […]

suppose there is a quadrilateral $ABCD$. any point $P$ which lies inside the quadrilateral satisfies $PA^2+PC^2 = PB^2+PD^2$. Should such a condition exist always in a rectangle or a square?.can there be any other quadrilateral in which such a point exists.

Given a surface with area $A$, what is the expected area of the region occupied by $k$ possibly overlapping random circles with equal radii $r$? For example, I would like to estimate the area of the black region of this simulation. If it helps, we can assume the edges of surface wraps around, so the […]

I’ve a doubt about compact manifolds and orientability. I know that Compact Manifolds in $\mathbb{R^3}$ are orientable. My questions is: The statement above is valid only for compact manifolds without boundary (in this case, closed manifolds)? I’m asking this because I’d read that the Möbius-Strip with its boundary is a compact manifold. Can someone explain […]

While solving a book on ellipses, I came across the following property of an ellipse which was given without proof :- If the normals be drawn at the extremities of a focal chord of an ellipse, a line through their point of intersection parallel to the major axis will bisect the chord. Working with the […]

I have two polygons $P_1 \subseteq P_2$ in the plane, and I would like to determine to which polygon a given point $p\in P_1 \setminus P_2$ is “closest”, by means of the following measure: $$r(p)=d(p,P_1)/d_p(P_1,P_2),$$ where $d(p,P_1)$ is the distance between $p$ and $P_1$ and $d_p(P_1,P_2)$ is the distance between $P_1$ and $P_2$. The $p$ […]

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Mean value theorem on Riemannian manifold?
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Sobolev inequality
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Axiom Systems and Formal Systems
Step Function and Simple Functions