A compass is essential for drawing arcs and circles, which are fundamental operations in geometric constructions. Other tools like a ruler or protractor are used for different tasks.

A straightedge is used to draw precise straight lines, which is a basic skill in geometric constructions. It does not measure angles or draw circles.

Drawing an arc from the vertex that intersects both sides is the essential first step in constructing an angle bisector. This step lays the groundwork for marking equal distances on each side of the angle.

The compass is crucial for transferring distances accurately by setting a specific radius. This method ensures the replication of lengths in various constructions.

An equilateral triangle is constructed by drawing congruent arcs from each vertex, ensuring that all sides are equal. This method relies on the properties of circles and their intersections.

Every point on a perpendicular bisector is equidistant from the two endpoints of the segment. This key property is used in determining centers of circles and triangles.

A tangent line must be perpendicular to the radius at the point of contact. This perpendicularity is the foundational property used in constructing tangent lines accurately.

An equilateral triangle has all internal angles equal to 60 degrees, providing an effective way to construct a 60-degree angle with basic tools. This method is straightforward and widely used in geometric constructions.

The intersection of two arcs drawn with equal radii from different centers yields a point that is equidistant from both centers. This concept is critical when constructing triangles and circles.

The property of alternate interior angles being equal when lines are parallel is used to construct a parallel line through an external point. This method helps ensure the new line maintains the required angular relationship to the given line.

The correct method involves drawing arcs from the given point that intersect the line at two distinct points. This helps in constructing a perpendicular line by determining equidistant points along the line.

Dividing a segment into equal parts is effectively accomplished by constructing similar triangles through the use of parallel lines. This method relies on proportional relationships between the segments.

Constructing a square requires all sides to be equal in length and every angle to be a right angle. Drawing perpendicular lines and ensuring equal segments is the cornerstone of this construction.

The perpendicular bisectors of chords in a circle intersect at the center of the circle. This principle is fundamental for locating the center accurately in geometric constructions.

To reflect a point across a line, you draw a perpendicular from the point to the line and then mark a point on the opposite side at the same distance. This technique ensures the reflected point is equidistant from the line as the original point.

The incenter, which is the center of the inscribed circle, is found at the intersection of the angle bisectors. This point is equidistant from all sides of the triangle.

A common method to draw a parallel line through an external point involves constructing a transversal to create corresponding or alternate interior angles equal to those of the given line. This guarantees that the new line is parallel.

The circumcenter is located at the intersection of the perpendicular bisectors of the triangle's sides. This point is equidistant from all three vertices, making it the center of the circumscribed circle.

A tangent line is defined by its perpendicular relationship to the radius at the point of tangency. This property is the cornerstone of constructing an accurate tangent from an external point.

The centroid is the point where all the medians of a triangle intersect. It divides each median into segments with a 2:1 ratio, with the longer portion adjacent to the vertex.