Regular Polygons Inscribed in a Circle - Geometry

Angle bisector of $\angle ABC$. Divides the angle into two equal parts.

Every point on it is equidistant from $A$ and $B$. Property of the perpendicular bisector construction.

Points on it are equidistant from the two sides of the angle. Property of the angle bisector construction.

The distance between the two arc intersection points on the original angle. This distance determines the second arc's radius.

At two points on opposite sides of $\overline{AB}$. Ensures the perpendicular bisector can be drawn.

Use radius greater than $\frac{1}{2}AB$ so the arcs intersect. Radius must exceed half the segment length for intersection.

Transfer length using a compass, not a ruler measurement. Compass preserves exact distances, not ruler measurements.

Keep the compass width fixed at $AB$ until marking the copied point. Changing width loses the original segment length.

The angle bisector must be a ray starting at vertex $B$. Angle bisectors originate from the angle's vertex.

Draw line $\overline{XY}$. Connect intersection points to form the perpendicular bisector.

With the same radius, draw arcs centered at $D$ and $E$ to intersect. Creates the angle bisector through arc intersection.

Transfer the chord length between arc hits to locate the second ray. Completes the angle copying process.

A line through $P$ parallel to $\ell$. Congruent corresponding angles create parallel lines.

The constructed line is perpendicular to $\ell$. Right angles define perpendicular lines.

Perpendicular bisector of $\overline{AB}$. A line through the midpoint, perpendicular to the segment.

If corresponding angles are congruent, then the lines are parallel. Congruent corresponding angles guarantee parallel lines.

To draw lines and rays, not to measure distances. Only for drawing straight lines, not measuring.

To transfer distances and create arcs or circles. Transfers distances and draws circular arcs.

Point $M$ on $\overline{AB}$ with $AM = MB$. Divides the segment into two equal parts.

A ray from $B$ that splits $\angle ABC$ into two congruent angles. Creates two angles of equal measure from the original angle.

Mark point $Q$ where the arc meets the ray so $PQ = AB$. Completes the segment copy with the correct length.

$AB = CD$. Segments with identical lengths are congruent.

$m\angle 1 = m\angle 2$. Angles with identical measures are congruent.

The compass width equal to the original segment length. Maintains the distance needed for accurate segment transfer.

Draw a ray with endpoint $P$. Provides the starting point and direction for the copied segment.

Draw a ray $\overrightarrow{PX}$ to be one side of the new angle. Establishes one side of the angle to be constructed.

A compass. Creates circular arcs needed for geometric constructions.

Use equal-radius arcs from $A$ and $B$ that intersect. Equal arcs ensure the bisector is equidistant from endpoints.

Compass radius must be greater than $\frac{1}{2}AB$. Ensures arcs intersect on both sides of the segment.