Flat Geometry

Notable points of the triangle

In the previous post we talked about Thales’ theorem. Now, let’s talk notable points of the triangle. History The term “ceviana” comes from the name of the Italian mathematician Giovanni Ceva. A cevian is a line segment that connects a vertex of a triangle to any point on the line that contains the opposite side. The notable points of the triangle came from studies of Cevians. Examples of cevians include the median, height, and bisector. The median, for example, connects the vertex from which it starts to the midpoint of the side opposite that vertex. The barycenter is the meeting point of the medians and represents the center of gravity of the triangle.

June 25, 2024 Read

Rectangle, Rhombus and Square

In this post the central point of the theme is the Rectangle, Rhombus and Square, as well as their properties. The history of squares and rectangles dates back to antiquity, when civilizations such as the Egyptian and Mesopotamian already used these geometric shapes to measure land, build monuments and symbolize religious concepts. Introduction Notable quadrilaterals are geometric figures that play a fundamental role in mathematics. Let’s explore its historical origin and properties.

June 18, 2024 Read

Parallelogram

In this post the central point of the topic is Parallelograms, as well as their properties. A parallelogram is a quadrilateral with four sides, where opposite sides are parallel. Introduction Every parallelogram has rectangle, rhombus and square as subsets. Definition: A convex flat quadrilateral is a parallelogram if, and only if, its opposite sides are parallel. $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$ are parallel. General concepts of the Parallelogram Opposite sides of a parallelogram are congruent, that is, they have the same measurement. Opposite angles of the parallelogram are congruent (same measurement). The consecutive angles of the parallelogram are supplementary, whose sum totals 180°. The sum of internal or external angles adds up to 360°. The diagonals of the parallelogram intersect at their respective midpoints (middle of the figure). A little of history The history of the parallelogram in mathematics is intrinsically linked to the evolution of geometry. The parallelogram, like a quadrilateral with opposite sides parallel, is a fundamental figure in plane geometry.

June 14, 2024 Read

Trapeze

In this post the central point of the topic is trapezoids, as well as their properties. A trapezoid is a flat geometric figure formed by four sides. Two of them are parallel and are called bases. The geometry of the trapezoid has its roots in Ancient Greece, where mathematicians such as Euclid and Archimedes studied its properties1. The name “trapezium” comes from the Greek trapezion, which means “square table”. The trapeze was also used by the Egyptians in the construction of pyramids and temples.

June 7, 2024 Read

Introduction to Quadrilaterals

In this post, the central point of the topic is quadrilaterals, which are the foundations for beginning the study of geometry. The history of quadrilaterals dates back to ancient times, with the famous Greek mathematician Euclid being one of the first to study their properties. The pioneering geometers in the study of quadrilaterals were Thales of Miletus (c. 624 – 546 BC), Pythagoras of Samos (c. 585 – 500 BC), Euclid of Alexandria (c. 365 – 300 BC).

June 6, 2024 Read

Tales' Theorem

In the previous post we talked about similarity of triangles. Now, let’s talk about Thales’ Theorem, which are fundamental concepts in plane geometry, especially in triangles. Story of Thales of Miletus Thales of Miletus was an important pre-Socratic Greek thinker, philosopher and mathematician. He is considered by some to be the “Father of Science” and “Western Philosophy”. His main ideas expanded theoretical horizons in the areas of mathematics, philosophy and astronomy. Let’s explore more about his life and contributions.

June 5, 2024 Read

Similarity of Triangles

In the previous post we talked about perpendicularity. Now, let’s talk about Similarity of Triangles, which are fundamental concepts in plane geometry, especially triangles. History of the similarity of triangles The optics used in entrance exams were discovered by Pierre Fermat in his work Variar Opera Mathematica of 1679. His original name is Clement Samuel Fermat, but he became known as Pierre Fermat. From Fermat’s studies we highlight two important results:

May 31, 2024 Read

Perpendicularity

In the previous post we talked about parallelism of triangles. Now, let’s talk about perpendicularity, which are fundamental concepts in plane geometry, especially in triangles. Perpendicularity is a fundamental concept in geometry that describes the relationship between two lines or straight segments that meet forming an angle of 90 degrees. These lines are called perpendiculars. History of perpendicularity Although there is no single specific mathematician credited with discovering perpendicularity, several ancient mathematicians and cultures contributed to the development of this concept, such as:

May 29, 2024 Read

Parallelism

In the previous post we talked about congruence of triangles. Now, let’s talk about parallelism, which are fundamental concepts in plane geometry, especially in triangles. Parallelism in geometry is a fundamental concept that describes the relationship between lines and planes. The History of Parallelism in Geometry The concept of parallelism in geometry has its roots in antiquity, specifically with the Greek mathematician Euclid, around 300 BC. He introduced the parallel postulate, which states that a single straight line passes through a point parallel to a given straight line. This postulate is one of the bases of Euclidean Geometry, which is the geometry we study on a daily basis.

May 16, 2024 Read

Congruence of Triangles

In the previous post we talked about initial concepts about triangles. Now, let’s talk about properties, specifically about the congruence of triangles. History The congruence of triangles is a fundamental concept in geometry, and its discovery dates back many centuries ago. Let’s explore a little of this story: Euclid and the Elements: The Greek mathematician Euclid, around 300 BC, formalized the principles of geometry in his famous book “The Elements”. In this work, he established the criteria for the congruence of triangles, as LLL - Side Side Side, LAL - Side Angle Side, ALA - Angle Side Angle and LAAo - Side Angle Opposite angle. These criteria are known as “congruence postulates”.

May 13, 2024 Read

Introduction to Triangles

A triangle is a three-sided polygon, in which we use straight segments for the sides. It is a particular case of quadrilaterals, as it is a quadrilateral divided by 2. Definition: Given three non-collinear points A, B and C1, the meeting of the segments $\overline{AB}$, $\overline{AC}$ and $\overline{BC}$ is called a triangle. Basic elements Vertices: can be defined as meeting points between the sides of a triangle. Generally speaking, vertices are meeting points between sides of any polygon. From the previous figure, points A, B and C are vertices of the triangle ($\Delta ABC$) In particular, a triangle is formed by 3 vertices.

May 10, 2024 Read