Circle Constructions

Geometric constructions involve drawing shapes, angles, or lines accurately using only a compass and a straightedge (an unmarked ruler). When working with circles, these tools allow us to create precise tangents, inscribe and circumscribe circles around triangles, and draw regular polygons.

Constructing a Tangent from an External Point

To construct a tangent line to a circle with center $O$ from an external point $P$:

1. Connect the points: Draw a line segment connecting the center $O$ and the external point $P$ to form segment $OP$.
2. Find the midpoint: Construct the perpendicular bisector of $OP$ to find its midpoint. Let's call this midpoint $M$.
3. Draw a second circle: Place the compass point on $M$ and set the width to $MO$ (which is exactly half the length of $OP$). Draw a new circle.
4. Mark the intersections: This new circle will intersect the original circle $O$ at exactly two points. Let's call them $A$ and $B$.
5. Draw the tangents: Use the straightedge to draw a line from $P$ through $A$, and another line from $P$ through $B$. These two lines are the required tangent lines to circle $O$.

Inscribed and Circumscribed Circles of a Triangle

Circumscribed Circle (Circumcircle)

A circumcircle is a circle that passes through all three vertices of a triangle.

1. Construct the perpendicular bisectors of at least two sides of the triangle.
2. The point where these bisectors intersect is called the circumcenter.
3. Place the compass point on the circumcenter, set the pencil to touch any vertex of the triangle, and draw the circle.

Inscribed Circle (Incircle)

An incircle is a circle inside the triangle that is perfectly tangent to all three sides.

1. Construct the angle bisectors of at least two angles of the triangle.
2. The point where these bisectors intersect is called the incenter.
3. Construct a perpendicular line from the incenter to one of the triangle's sides to find the exact radius of the incircle.
4. Place the compass point on the incenter, set it to this perpendicular radius, and draw the circle.

Inscribing a Regular Hexagon in a Circle

A regular hexagon is one of the easiest polygons to construct because its side length is exactly equal to the radius of its circumscribed circle.

1. Draw a circle with center $O$ and keep your compass set to the circle's radius, $r$.
2. Mark a starting point anywhere on the circle's circumference.
3. Place the compass point on this mark and draw an arc that intersects the circle's edge.
4. Move the compass point to this new intersection and draw another arc further along the circle.
5. Repeat this process until you have made 6 marks around the circle. (The 6th mark will land perfectly back on your starting point).
6. Use a straightedge to connect the adjacent marks with straight lines. You have successfully inscribed a regular hexagon!