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Foundations of Geometry: Exploring Basic Constructions and their Applications

Jihao Jen^{* *}Correspondence: Jihao Jen, Department of Mathematical Sciences, Nanjing University, Jiangsu, China, Email:

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Description

Geometric constructions refer to the process of drawing accurate and precise figures using a compass and a straightedge, without the aid of measurement tools such as rulers or protractors. These constructions have been used for thousands of years to create various geometric shapes, and they continue to be important in both mathematics and engineering.

One of the earliest known geometric constructions was the process of drawing a circle using a compass. The ancient Greeks were experts in this technique, and they used it to study various geometric shapes and angles. For example, they discovered that the ratio of the circumference of a circle to its diameter is always constant, a value that we know today as pi.

Another famous geometric construction is the process of constructing an equilateral triangle. This involves drawing three congruent line segments that form a triangle with three equal angles of 60 degrees each. This construction is important in many areas of mathematics and engineering, and it can be used to create other geometric shapes, such as hexagons and dodecagons.

Other important geometric constructions include drawing perpendicular bisectors, which can be used to find the midpoint of a line segment, and drawing angle bisectors, which can be used to find the angle bisector of a triangle. These constructions are essential for many geometric proofs and are also used in realworld applications, such as architecture and construction.

One of the most interesting aspects of geometric constructions is that they can be used to solve complex mathematical problems that would be difficult or impossible to solve otherwise. For example, geometric constructions can be used to find the square root of a number or to solve quadratic equations. They can also be used to create various geometric patterns, such as tessellations and fractals, which have important applications in art and design.

Basic constructions in geometry

Constructing a line segment: To construct a line segment, place the straightedge between two points and draw a straight line between them.

Constructing a perpendicular bisector: To construct a perpendicular bisector, place the compass on one end of the line segment and draw an arc that intersects the line segment. Then, place the compass on the other end of the line segment and draw another arc that intersects the first arc. Draw a straight line through the intersection of the two arcs, and this line will be the perpendicular bisector of the line segment.

Constructing an angle: To construct an angle, place the compass on the vertex of the angle and draw an arc that intersects both sides of the angle. Then, draw a straight line from the vertex to the intersection of the two arcs.

Constructing an equilateral triangle: To construct an equilateral triangle, first draw a line segment. Then, place the compass on one end of the line segment and draw an arc. Move the compass to the other end of the line segment and draw another arc that intersects the first arc. Draw a straight line through the intersection of the two arcs and the midpoint of the line segment. This will create an equilateral triangle.

Constructing a square: To construct a square, first construct a line segment. Then, construct a perpendicular bisector of the line segment. Place the compass on one end of the line segment and draw an arc that intersects the perpendicular bisector. Move the compass to the other end of the line segment and draw another arc that intersects the perpendicular bisector. Draw a straight line through the two intersection points of the perpendicular bisector and the arcs. Then, draw two more lines perpendicular to this line and through the endpoints of the line segment. This will create a square.