## Sekai ๐ ๐บ

Sekai (ไธ็) is the kanji for “the world”. That’s a great word because of the scale that it designates.

Background I’ve written about bringing user-defined KaTeX macros into Hugo a few years ago. Looking back, I realized that I only knew how to copy code at that time.
Goal To replace the current code for my math editor with Showdown-KaTeX. To bring mhchem into my LaTeX + Markdown sandbox. Motivation To get the benefits of the three free (as in “freedom”) technologies.
Markdown syntax is (much) simple(r than its LaTeX equivalent, especially for tables, ordered/unordered lists, etc).
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## Exponential Function Product Rule

### A first definition of exponential and logarithmic functions

Motivation from compound interest Increase the number of times that the interest is compounded each year ($n$), so as to increase the final amount of money ($A$)
$$A = P \left(1 + \frac{r}{n}\right)^{nt}.$$ $t$ and $r$ are the number of years and the interest rate per annum.
As $n$ becomes large, we can approximate the amount by
$$A = P e^{rt}.$$ The value of $A$ in the above formula is the amount compounded continuously.
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## Existence of Four Triangle Centers

### A vector proof

Settings Let
$P$ be an arbitrary reference point. $\triangle ABC$ be a triangle. $\vec{a} = \overrightarrow{PA}$, $\vec{b} = \overrightarrow{PB}$, $\vec{c} = \overrightarrow{PC}$ Remark: $O$ is reserved for circumcenter.
symbol name meaning $G$ centroid center of gravity $H$ orthocenter three “heights” are concurrent $I$ incenter center of inscribed circle $O$ circumcenter center of circumscribed circle Centroid Verify that $(\vec{a} + \vec{b} + \vec{c})/3$ satisfy the constraints.
Orthocenter Let
$H$ be the point of intersection of two altitudes $AA_H$ and $BB_H$ $\vec{h} = \overrightarrow{PH}$ \begin{align} (\vec{h} - \vec{a}) \cdot (\vec{b} - \vec{c}) &= 0 \\ (\vec{h} - \vec{b}) \cdot (\vec{c} - \vec{a}) &= 0 \end{align} Add these two equations together.
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## Katex Boxed Equations

Goal To render a bordered box with multiple equations.
Background My previous article about dot products contains some boxed math expressions: some are inline and some take up the whole line.
For inline math, that’s simple: either $\fbox{text}$ or $\boxed{a=1}$ will do. For displayed math, these two KaTeX commands will ignore the newline command \\.
Solution A simple Google search ‘“katex” AND “box”’ saved the day. It led me to this relevant Stack Overflow question, whose accepted answer suggested the use of custom CSS.
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## Tikz to SVG in VS Code

### My little LaTeX Workshop recipe

Goal View SVG graphics generated from TikZ code in VS Code.
Compile TikZ code and preview SVG in VS Code with LaTeX Workship and SVG Preview
Part I: generate SVG using LaTeX Workshop From the IDE’s parameters tab, I found the tools and recipes parameters. I clicked my user settings settings.json twice: once through the tools, and once through the recipes. Each click generated an array of default recipes or tools.
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## Dot Products

### My complicated derivation

Preface There’s a simpler derivation using
the geometric defintion linearity from the definition orthogonality of the canonical basic vectors $\vec{i}$, $\vec{j}$ and $\vec{k}$. If I had read that, I wouldn’t have type this document in LaTeX. The following content was transcribed from a PDF file that I made three days ago.
Content Recall: $\sum_{i=m}^{n} a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n},$ where $m$, $n$ are integers.
Example: $\sum_{i=3}^6 i^2 = 3^2 + 4^2 + 5^2 + 6^2 = 86$
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## Functions for Arts

### Summary of my two-day work

my HTML slides You may view the source code of this blog to see the source files. Goal To provide an introduction to the formal definition of functions for arts stream students.
I aim to
minimize the calculations use daily-life examples favor pictures over text. IT skills used Inkscape C-S-a for alignment. C-S-f for boundary and colors, and line patterns (i.e. arrows). C-S-d for document size. clipping process: prepare the object to be clipped.
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## dvisvgm Guide for TikZ Pictures with Pattern Filling

### A quick note of the right commands for the conversion

Problem I had issues in getting a SVG from a TikZ picture containing a pattern filling: https://github.com/mgieseki/dvisvgm/issues/158. The reported issue was created last year, and it’s marked as fixed. Nonetheless, I had this issue even though my MikTeX was up-to-date.
Discussion I wanted to confirm whether my dvisvgm.exe, which had been shipped with MikTeX, was latest. I first asked at https://github.com/mgieseki/dvisvgm/issues/184, but I was referred to MikTeX’s GitHub repository since it’s about the MikTeX-packaged version of this tool.
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## Distance From a Line

### TikZ figures for a derivation from dot product

Background A secondary school student posted a coordinate geometry question on a Discord homework help server. A helper replied to that question with an illustration of the Perpendicular Distance Formula
$$ d((x_1,y_1), L) = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}, $$ where $L$ is the line $Ax + By + C = 0$ without proof.
Goal To provide an illustrated derivation of this formula.
Recall A basic property of the dot product: the “algebraic definition” is equivalent to the “geometric definition”.
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## Elementary Log Inequality

### TikZ figures for some graphical proofs of this inequality

Goal Show that
$$ \ln (x) \le x - 1 $$ for all $x > 0$.
Proof by inverse functions Think about their reflection along the line $y = x$ (i.e. their inverse function). Then we get
$$e^x \ge x + 1$$ for all $x \in \Bbb{R}$, which is obviously true.
Proof by definite integrals Here’s a second proof using definite integrals.
Case 1: x > 1 The rectangle $[1,x] \times [0,1]$ contains the region under the graph of the reciprocal $y = 1/x$ in the domain $[1,x]$.
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