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), […]

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 […]

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 […]

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 […]

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 […]

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 […]

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.

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 $m<n$)?

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 […]

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 […]

Intereting Posts

The area of the superellipse
Definition of a universal cover and the universal cover of a point
Connectedness of the spectrum of a tensor product.
Why isn't $\mathbb{CP}^2$ a covering space for any other manifold?
Sign of det(UV) in SVD
A simple question about open set
S4/V4 isomorphic to S3 – Understanding Attached Tables
circular reasoning in proving $\frac{\sin x}x\to1,x\to0$
Convergence in measure implies convergence almost everywhere of a subsequence
Correct process for proof in graph theory.
Coefficient in the Fourier expansion of the cusp form
Convergence of $\sum_{k=1}^{n} f(k) – \int_{1}^{n} f(x) dx$
Proof that gradient is orthogonal to level set
$\lim_{x\to 1^-} \sum_{k=0}^\infty \left( x^{k^2}-x^{(k+\alpha)^2}\right)$
Show that for any $w \in \mathbb{C}$ there exists a sequence $z_n$ s.t. $f(z_n) \rightarrow w$