## The Mathematical Symbol "Homothetic (∻)"

The ∻ Symbol in Mathematics: Denoting Homothety

The mathematical symbol ∻ stands for "Homothetic", often used in geometry to represent the concept of homothety. A homothety (or central dilation) is a transformation of the plane that multiplies all distances by a common factor from a fixed point called the center of homothety. In this article, we will explore the primary usages of homothety and provide two illustrative examples.

## Definition

Homothety, denoted by the ∻ symbol, is a transformation that fixes a point $$C$$ (the center) and maps any other point $$P$$ to $$P'$$ such that the segment $$CP'$$ is a scalar multiple $$k$$ of segment $$CP$$. The scalar $$k$$ is known as the ratio of homothety.

## Examples

• Example 1: Let's consider a circle with center $$O$$ and a point $$P$$ on its circumference. If we perform a homothety with center $$O$$ and ratio $$k=2$$, then the new position of $$P$$ (let's call it $$P'$$) will lie on a circle with twice the radius and the same center $$O$$.
• Example 2: Consider a triangle $$ABC$$ in the plane. If we perform a homothety with a center outside the triangle, say point $$D$$, and a ratio $$k$$, then the triangle will either expand or contract (based on the value of $$k$$), with all vertices moving along the lines $$AD$$, $$BD$$, and $$CD$$.

In summary, homothety is a central concept in geometry, especially when studying similar figures. The ∻ symbol provides a standardized notation to denote this transformation, enabling mathematicians and students alike to explore and describe geometric properties and relationships with clarity. ## Are You Good at Mathematical Symbols?

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## Codes for the ∻ Symbol

 The Symbol ∻ Alt Code Alt 8763 HTML Code ∻ HTML Entity ∻ CSS Code \223B Hex Code ∻ Unicode U+223B

## How To Insert the ∻ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ∻ symbol is to copy and paste it into your document.

Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8763. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

(Method 3) Use the HTML Decimal Code (for webpages).

HTML TextOutput
<b>My symbol: &#8763;</b>My symbol: ∻

(Method 4) Use the HTML Entity Code (for webpages).

HTML TextOutput
<b>My symbol: &homtht;</b>My symbol: ∻

(Method 5) Use the CSS Code (for webpages).

CSS and HTML TextOutput
<style>
span:after {
content: "\223B";}
</style>
<span>My symbol:</span>
My symbol: ∻

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML TextOutput
<b>My symbol: &#x223B;</b>My symbol: ∻
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x223B to place the ∻ symbol on your canvas. For example:
JavaScript Text
const x = "0x"+"E9"
ctx.fillText(String.fromCodePoint(x), 5, 5);
Output

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ∻ is U+223B. The important part is the hexadecimal number after the U+, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:
TypeOutput
223B
[Hold down Alt]
[Press x]

(The 223B turns into ∻. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u223B. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u223B"
document.write("My symbol: " + str)
My symbol: ∻