Practical Geometry - Construction of a Circle

A circle is a simple closed curve where every point on the boundary is at an equal distance from a fixed point called the center. Visually, if you place a pin at a point $O$ and rotate a pencil around it at a fixed distance, the path traced is a circle.

The radius is the constant distance from the center of the circle to any point on its boundary, usually denoted by $r$. Visually, it is a straight line segment like the spoke of a wheel connecting the middle to the outer rim.

The diameter is a line segment that passes through the center of the circle and has its endpoints on the circle's boundary, denoted by $d$. Visually, it is the longest possible straight line you can draw across the circle, effectively cutting it into two equal halves (semicircles).

To construct a circle using a compass, the metal pointer is placed firmly at the center point $O$, while the pencil lead is adjusted to the required radius. As you rotate the compass $360^{\circ}$, the pencil traces the circumference while the pointer remains stationary.

Points related to a circle can be categorized into three regions: the interior (points inside the circle), the boundary (points on the circle), and the exterior (points outside the circle). Visually, if a point $P$ is at distance $x$ from the center $O$, it is in the interior if $x < r$, on the boundary if $x = r$, and in the exterior if $x > r$.

A chord is any line segment joining two points on the boundary of the circle. While many chords can be drawn, the diameter is unique because it is the only chord that passes through the center $O$. Visually, chords of different lengths can be seen 'criss-crossing' the circle's interior.

When constructing circles with the same center but different radii, they are called concentric circles. Visually, this looks like a 'bullseye' target or ripples in a pond where multiple circular rings share a single middle point.