📕 The Circle and Its Properties

📗 The Equation of a Circle

A circle is the set of all points $(x, y)$ in a plane that are at a fixed distance (the radius, $r$) from a given point (the center, $(h, k)$). The equation of a circle is derived from the distance formula:

For radius r and center $C(h, k)$.

\color{#FFFF11}  
(x - h)^2 + (y - k)^2 = r^2

By definition, the circumference is the set of all points $P(x, y)$ at a distance from the center $[C(h, k)]$ of $r$ unit. This is: $d(P, C) = r$.

Expanding this equation, we obtain the general equation of a circle:

\color{#FFFF00}  
𝑥²＋𝑦²＋𝐴𝑥＋𝐵𝑦＋𝐶＝０

where the parameters relate to the center and radius by:

\color{#FFEE00}  
𝐴 = -2𝑎 

\color{#FFDD00}  
𝐵 = -2𝑏

\color{#FFCC00}  
𝐶 = 𝑎²+𝑏²-𝚛²

\color{#FFBB00}  
𝚛² = \frac{𝐴²+𝐵²-4𝐶}{4}

\color{#FFAE03}  
(x-h)²+(y-k)² = r²

\color{#E67F0D}  
x² - 2hx + h² + y² - 2ky + k² = r²

\color{#FE4E00}  
x² + Ax + y² + By + C = 0

📘 Arc of a Circle

An arc is a connected portion of a circle. It is usually specified by its endpoints and its length, which is measured as an angle subtended at the center of the circle.

📙 Angle and Its Types

An angle is the space formed by two rays sharing a common endpoint, called the vertex.

📖 Angle of Elevation & Depression

1. Angle of Elevation (∠⦨):

   The angle between the horizontal line and an upward line of sight.

2. Angle of Depression (⦧⦫):

   The angle between the horizontal line and a downward line of sight.

\text{Elevation:} \quad
\begin{array}{c}
       \alpha^\circ \underline{  \nearrow } \\

\end{array}

\text{Depression:} \quad
\begin{array}{c}
       \alpha^\circ \overline{  \searrow } \\

\end{array}

📗 Bearings

A bearing is the angle measured clockwise from the North direction. It is commonly used in navigation.

\begin{array}{c}
ℕ \\
↑ ⦬
\end{array}

📕 Adjacent Elements in Geometry

1. Adjacent Sides

   Two sides that share a common vertex or if they share an angle.

A┌───────┐C   
B└───────┘D   
AC and AB are adjacent because they share the point A 

2. Adjacent Angles

   Two angles that share a common side and a vertex.

A      B             
⟍      /
  ⟍   /  
    ⟍/________  C
     O
∠AOB is adjacent to ∠BOC because they share the line →OB

📗 Angle Sum Theorem

The sum of interior angles in polygons:

3. Triangle

   • $𝑎°﹢𝑏°﹢𝑐°﹦ 𝟷𝟾𝟶° = 𝜋 ㎭$

4. Quadrilateral

   • $𝑎°﹢𝑏°﹢𝑐°﹢𝑑°﹦ 𝟹𝟼𝟶° = 𝟤𝜋 ㎭$

n. Any Polygon:

• $number⠀of⠀vertices⠀⨯⠀𝟷𝟾𝟶°⠀-⠀𝟹𝟼𝟶°$
• $(number⠀of⠀vertices -2)⠀⨯⠀𝟷𝟾𝟶°$

📙 Annulus – The Region Between Two Concentric Circles

An annulus is the region between two concentric circles of radii $𝚁$ and $𝚛$.

• ◎ $A=𝜋(𝚁²-𝚛²)$