## Line Equation in Intercept Form

### LaTeX code for a TikZ figure

This is a first draft of a TikZ picture illustraing this classical formula to be used for math help channels. Adding \caption{for the picture} without “Figure 1:” requires \usepackage{caption} and wrapping with \begin{figure}. It also possible to use the primitive TeX command \par, but it would be complicated to use that with standalone. In the previous post, the SVG picture from the LaTeX table in an article has too much useless whitespace around the table. [Read More]

## Second Homothety between Nine-Point Circle and Circumcircle

### Proof of Euler line by h(G, −1/2)

Motivation I saw someone illustrating his/her solution with a “superior triangle”. This reminds me the homothety about the centroid of factor −1/2. The above picture \usetikzlibrary{calc} for computing coordinates from those of existing points. (A)!.25!(B) means $(A)+.25[(B)-(A)]$. \begin{tikzpicture}[scale=2] \coordinate (D) at (-0.7,1); \coordinate (E) at (-1,0); \coordinate (F) at (1,0); \coordinate (A) at ($(E)!.5!(F)$); \coordinate (B) at ($(F)!.5!(D)$); \coordinate (C) at ($(D)!.5!(E)$); \coordinate (G) at ($(D)!.5!(E)!1/3!(F)$); \draw (A) -- (B) -- (C) -- cycle; \draw (D) -- (E) -- (F) -- cycle; \begin{scriptsize} \fill (G) circle (0. [Read More]

## Basic Symmetries in Nine-Point Circle

Motivation Someone on Discord asked about the existence of the nine-point circle. It’s well-known that that can be proved by homothety. Little reminder about homothety Homothety preserves angles (and thus parallel lines). Homothetic polygons are similar, so the ratio of the corresponding sides is the same. Considering the radii of a circle under a homothety, we see that a homothety maps a circle to another circle. Notation H: orthocenter G: centroid O: circumcenter ω: circumcircle HA: feet of altitude with respect to A. [Read More]

## To Be Improved Normal Curve

### First function plot with pattern fill

Adaptations for standalone documents:

• \usetikzlibrary{pattern} before \begin{document}
• \pgfplotsset{compat=1.6}
\begin{tikzpicture}
\begin{axis} [axis lines=center,legend style={at={(0.7,0.7)},anchor=south west}]
\addplot [domain=-3:3, thick, smooth, yellow] { 1/sqrt(2*pi)*exp(-x^2/2) };
\addlegendentry{$y = \tfrac{1}{\sqrt{2\pi}} e^{-x^2/2}$};
\addplot [dashed, yellow] coordinates {(1.5,0) (1.5,0.14)};
\addplot[domain=-3:1.5, pattern=north east lines,draw=none, fill opacity=0.3]
{ 1/sqrt(2*pi)*exp(-x^2/2) } \closedcycle;
\end{axis}
\end{tikzpicture}


## Trigonometric Functions by Unit Circle

For secondary school students, I define cosine and sine as the x and y-components of the point A (cos θ, sin θ) on the unit circle x² + y² = 1, and the tangent function as the quotient of sine over cosine. \begin{tikzpicture}[scale=3] \coordinate (O) at (0,0); \coordinate (H) at (0.6,0); \coordinate (A) at (0.6,0.8); \coordinate (E) at (1,0); \coordinate (T) at (1,0.8/0.6); \draw (O) circle (1); \draw[->] (-1.3,0) -- (1. [Read More]

## Matrix Diagonalisation and Change of Basis

Here’s the $\LaTeX$ code of my diagram for matrix diagonalisation to be used on Discord. Why do matrix diagonalisation on square matrix $P$? If we can find a diagonal matrix $D$ and a square matrix $Q$ such that $P = QDQ^{-1}$, then we can easily compute $(P + \lambda I)^n$ for any scalar $\lambda$ and integer $n$ because $D^n$ is easy to compute. $\begin{tikzcd} {{}} & {{}} & \cdots & {} \\ {{}} & {{}} & \cdots & {{}} \arrow["P", from=1-1, to=1-2] \arrow["{Q^{-1}}"', from=1-1, to=2-1] \arrow["D"', from=2-1, to=2-2] \arrow["Q"', from=2-2, to=1-2] \arrow["P", from=1-2, to=1-3] \arrow["D"', from=2-2, to=2-3] \arrow["P", from=1-3, to=1-4] \arrow["D"', from=2-3, to=2-4] \arrow["Q"', from=2-4, to=1-4] \end{tikzcd}$ After viewing the power of the Discord bot $\TeX{}$it, which renders $\LaTeX$ code on Discord, I gave up spending more time on exploring more functionalities of $\KaTeX$ (say, commutative diagrams) because Discord and $\LaTeX$ spread math knowledge much better than a static blog for basic math: the former allows instant feedback from the reader. [Read More]

## LaTeX Code for Linear System

\begin{align} A\mathbf{x} &= \mathbf{b} \\ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} &= \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \\ \begin{bmatrix} \vert & \vert & \vert \\ \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \\ \vert & \vert & \vert \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} &= \mathbf{b} \\ x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + x_3 \mathbf{a}_3 &= \mathbf{b} \tag{$\star$} \\ \begin{pmatrix} a_{11} x_1 + a_{12} x_2 + a_{13} x_3 \\ a_{21} x_1 + a_{22} x_2 + a_{23} x_3 \\ a_{31} x_1 + a_{32} x_2 + a_{33} x_3 \end{pmatrix} &= \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \\ x_1 \begin{pmatrix} a_{11} \\ a_{21} \\ a_{31} \end{pmatrix} + x_2 \begin{pmatrix} a_{12} \\ a_{22} \\ a_{32} \end{pmatrix} + x_3 \begin{pmatrix} a_{13} \\ a_{32} \\ a_{33} \end{pmatrix} &= \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \end{align} Source code: [Read More]

## Polar Rose in Julia

Background I’m doing exercise 4.9 of Think Julia, which asks for a function for a polar rose using Luxor’s turtle graphics. Difficulties Work out the geometric structure of the family of polar roses. The key is to construct some auxiliary isoceles triangles and work out the angles between them. One sees that they are parametrized by two varaibles n and k. n: number of petals k: petal increment constraint: k ≠ n ÷ 2 Handle the case when gcd(n, k) > 1, i. [Read More]

## Deleted Question on Semi-Simple and Projective but not Injective Module

A backup of a deleted PSQ : https://math.stackexchange.com/q/3955443/290189 OP : irfanmat It has a detailed answer by Atticus Stonestrom. It’s pity that his post got deleted. As there’s no reason for undeletion, I’m posting it here so as to preserve the contents. Question body Is there a semi-simple and projective but not injective module? I will be glad if you help. Response(s) In the non-commutative case, the answer is yes. Consider $R$ the ring of upper triangular $2\times 2$ matrices over a field $F$, and denote by $e_{ij}$ the element of $R$ with the $ij$-th entry equal to $1$ and all other entries equal to $0$. [Read More]

## Another Math.SE Double Integral Using Polar Coordinate

I wanted to post the following answer to a question on double integral on Math.SE, but someone had submitted his work before I finished typing. As a result, I’m posting this on my personal blog. Let $r = \sqrt{x^2+4y^2}$ and $t = \begin{cases} \tan^{-1}(2y/x) &\text{ if } x > 0 \\ \pi/2 &\text{ if } x = 0. \end{cases}$ Then $\begin{cases} x &= r \cos t \\ y &= (r \sin t)/2 \end{cases}$ and $D = { (r,t) \mid r \ge 0, t \in [\pi/4, \pi/2] }$. [Read More]