Articles of geometry

Beautiful triangle problem

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. […]

Points on the circle

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?

Rolling ellipse on line – tangent and normal of roulette

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 […]

Distance between points in a convex set and outside of a convex set.

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 – […]

Reference books for highschool Algebra and Geometry?

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 […]

condition required for a quadrilateral $ABCD$ such that every point inside $ABCD$ satisfies $PA^2+PC^2 = PB^2+PD^2$

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.

Expected occupied area of a surface covered with possibly overlapping random shapes.

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 […]

Compact manifolds and orientability

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 […]

Property of ellipses involving normals at the endpoints of a focal chord and the midpoint of that chord

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 […]

Distances between points and polygons

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 $p$ […]