Articles of geometry

How to get the co-ordinates of scaled down polygon

Let say I have a polygon. I need to draw another polygon inside this polygon which is scaled-down. See this image, I need inner Polygon co-ordinates. Given that I have outer polygon co-ordinates and scaled down value.

Find this angle, in terms of variables

I have a geometry question. Take a look at this figure: It is 3 circles, symmetrically placed so that the arc length that is outside is equal for all sides. I tried to determine the angle that is tangent to two sides as they intersect (Theta in the drawing) in terms of $L$ (arc length) […]

Three Circles Meeting at One Point

We have three triples of points on the plane, that is, $X=\{x_1, x_2, x_3\}$, $Y=\{y_1, y_2, y_3 \}$, and $Z=\{z_1, z_2, z_3\}$, where $x_i, y_i, z_i$ are points on the plane. I was wondering if there is a simple (or not-so-simple) algebraic relation satisfied by the coordinates of the 9 points if we know that […]

Reflection of a curve around a slant line

a fifth-degree function: y = 80*x^5-225*x^4+350*x^3-300*x^2+150*x-20 (the green curve in the image) needs to be reflected/mirrored around the line y=55x-20 (the blue line) and I am only interested in the segment [0,1]. While there is plenty of content on the internet on how to reflect around the axes or vertical/horizontal lines, I have not found […]

the 8 queens conjecture: the next level (or dimension)

i know you have possibly seen my last problem on the 8 Queens conjecture, and it was answered by a person named Peter Kagey. he mentioned that: With regard to a $n×n×n$ chessboard, one could simply place $n$ queens at the “bottom board” of the cube, and use the $n*n$ configuration. This argument shows that […]

How to prove that $ 6ab=(a+b)(a+b+c)$ for triangle

There exists $\triangle{ABC}$ with $AB=c, BC=a, CA=b$. Let I be the incenter and G be the centroid of $\triangle{ABC}$. Assume that $GI$ perpendicular to $CI$. Prove that $6ab=(a+b)(a+b+c)$.

angle inside a chordal quadrilateral

I am trying to solve this problem concerning this chordal quadrilateral. I’m supposed to find out $\beta$. Help is really needed since I study for an exam. $\beta$ should be in dependency of the angle at $M_2$: $\beta= f(\text{angle at}\space M_2)$. In the left bottom corner should be the point A, I was already playing […]

Diameter of a triangle

If $T\subseteq \mathbb R^2$ is a generic plane triangle, I want to find its diameter $$d=\sup\{\lvert| x-y \rvert|: x,y\in T\}$$ Intuitively I think that $d$ is the length of the longest edge of $T$. How can I formally prove this?

Area of hyperbolic triangle definition

I found this question recently in my booklet on hyperbolic geometry asking a very simple question but I could not answer it: Why can we not define the area of a hyperbolic triangle as in the plane as half the product of the perpendicular and the base? I know the half plane model and the […]

The diameter of a convex hull.

I want to prove the following statement: Given $A\subset \mathbb{R}^n$ let $C(A)$ be its convex hull. Prove that $\text{diam }(A)=\text{diam }(C(A))$. I can suppose that $A$ is a bounded closed set and I know that if $x,y\in A$ are such that $d(x,y)=\text{diam }(A)$ then $x,y\in \partial A$. I tried proving that if $z,w\in \partial C(A)$ […]