Theorem of Congruent Arcs in a Circle and Regular Polygons
In a circle divided into three or more congruent arcs, you can draw an inscribed regular polygon by connecting the ends of each adjacent arc, and a circumscribed regular polygon by connecting the tangents to the circle.
The inverse theorem also holds.
When you inscribe a regular polygon with n sides in a circle or position a regular polygon so that each side touches the circle (circumscription), you divide the circle into n congruent arcs.
A Practical Example
A] Inscribed Regular Polygons
Take a circle and identify four points on the circle so that they are equidistant from each other.
This divides the circle into four congruent arcs (arcs of equal length).
$$ \overparen{AB} \cong \overparen{BC} \cong \overparen{CD} \cong \overparen{AD} $$
Now, connect each of these points with its adjacent points using line segments.
The points become the vertices of an inscribed polygon in the circle. In this case, it's a square.
Each interior angle of the square touches the circle, and each side of the square is a chord of the circle.
Inverse Theorem
Consider a perfectly symmetrical shape, like a square or an equilateral triangle.
Identify the center (O) of the regular polygon, which is the point equidistant (d) from the vertices of the polygon, located at the intersection of the bisectors.
Draw a circle with center O and radius equal to distance d.
Each vertex of the regular polygon touches the circle at a point, and these points of contact divide the circle into equal sections or congruent arcs.
For example, if the regular polygon has 5 sides (regular pentagon), the circle is divided into 5 equal arcs.
B] Circumscribed Regular Polygons
Take a circle and divide it into n=4 congruent arcs.
Draw tangent lines at the ends of arcs AB, BC, CD, and AD.
The four tangent lines intersect at four points outside the circle: E, F, G, H.
Connecting these points forms a circumscribed regular polygon. In this case, it's a square.
The sides of the square touch the circle at one point per side, circumscribing the circle.
Inverse Theorem
Now, draw a regular polygon, such as a square.
Draw the bisectors of each angle and identify the center (O) of the regular polygon.
Now draw a circle that touches the sides of the polygon without cutting through them.
The point of contact between each side of the polygon and the circle is the point of tangency.
The points of tangency divide the circle into four equal (congruent) arcs.
And so on.