Articles of geometry

Bisectors of a triangle meet at point.

Prove that internal angle bisectors of $\triangle ABC$ meet at a point. The problem is that I have to prove this using the locus of a straight line and its properties, I can’t use vectors. The proof I can think of is very simple but extremely tedious. Let the coordinate of triangle be $(0,0), (a,0), […]

Computing a point having distances from 2 points respectively

I want to compute a point $p$ which has distances $d1$ and $d2$ from points $q1$ and $q2$ respectively. As I wanted a general answer, I used Maxima inputting the following script , $solve($ $[sqrt((num1-x)^2+(num2-y)^2)=num3,$ $sqrt((num4-x)^2+(num5-y)^2)=num6]$ $,[x,y]);$ Where all variables of the format $num$_ were constants in my mind and $p=(x,y)$. I expected an output […]

Fitting data to a portion of an ellipse or conic section

Is there a straightforward algorithm for fitting data to an ellipse or other conic section? The data generally only approximately fits a portion of the ellipse. I am looking for something that doesn’t involve a complicated iterative search, since this has to run at interactive speeds for data sets on the order of 100s of […]

Find the shortest distance from the triangle with with vertices in $(1,1,0),(3,3,1),(6,1,0)$ to the point $(9,5,0)$.

I have to find the shortest distance from the triangle with with vertices in $(1,1,0),(3,3,1),(6,1,0)$ to the point $(9,5,0)$. I cannot figure out how to do this. There are three possible cases: the minimum lies inside the triangle, on an edge or in a vertex. But what is a reasonable way to understand in which […]

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we prove that a regular $n$-gon is constructible by ruler and compass if and only if the number $cos(2\pi/n)$ (that is, the […]

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let’s take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P \in \mathcal{A}_3 \text{ such that } \overrightarrow{AP} \in \langle u \rangle \}$. Now, what is not clear to me is: if we consider an Euclidean affine […]

How to calculate the two tangent points to a circle with radius R from two lines given by three points

I need to calculate the two tangent points of a circle with the radius $r$ and two lines given by three points $Q(x_0,y_0)$, $P(x_1,y_1)$ and $R(x_2,y_2)$. Sketch would explain the problem more. I need to find the tangent points $A(x_a,y_a)$ and $B(x_b,y_b)$. Note that the center of the circle is not given. Please help.

Random points on a sphere — expected angular distance

Suppose we randomly select $n>1$ points on a sphere (all independent and uniformly distributed). What is the expected angular distance from a point to its closest neighbor? What is the expected angular distance from a point to its $m^{\text{th}}$ closest neighbor (where $m<n$)?

Tesseract projection into $3D$

I found this: The tesseract is a four dimensional cube. It has 16 edge points $v=(a,b,c,d)$, with $a,b,c,d$ either equal to $+1$ or $-1$. Two points are connected, if their distance is $2$. Given a projection $P(x,y,z,w)=(x,y,z)$ from four dimensional space to three dimensional space, we can visualize the cube as an object in familiar […]

The definition of distance and how to prove the ruler postulate in Euclidean geometry

I have started to read some books about geometry. At the moment I’ve just started to read Hilbert’s axioms and also some elementary books for highschool. From the basic perspective of an axiomatic system, the elementary books, I mean not necessarily for highschool but also not for mathematicians, start with some axioms that are not […]