Flat Geometry

Stewart Relationship

In previous posts we talked about ptheorems for any triangles. The Stewart Relation, also known as Stewart’s Theorem, is an important formula in geometry that relates the lengths of the sides of a triangle to the length of a cevian. A cevian is a line segment that starts from a vertex of a triangle and meets the opposite side. This relationship is useful for solving problems involving determining the lengths of segments within a triangle, such as medians, bisectors, and heights.

July 12, 2024 Read

Theorem for any triangles

In previous posts we talked about point power. The Sine Theorem is a fundamental mathematical tool in trigonometry, which establishes an important relationship between the sides of a triangle and the sines of its angles. This theorem is applicable to any type of triangle, be it acute, obtuse or scalene, and is essential for solving problems involving the geometry of triangles. The Law of Sines states that, in any triangle, the ratio between the length of one side and the sine of the opposite angle is constant and equal for all three sides and angles of the triangle.

July 12, 2024 Read

Right triangle relations

The right triangle is a geometric figure that fascinates mathematicians and students due to its simultaneous simplicity and complexity. The metric relations of the right triangle are a set of formulas that relate the lengths of the sides and the projections of the legs on the hypotenuse. These relationships are fundamental to understanding more advanced concepts in trigonometry and analytical geometry. The best known of these relationships is the Pythagorean Theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, the legs. Mathematically, this is expressed as $a^2 = b^2 + c^2$, where $a$ is the hypotenuse and $b$ and $c$ are the legs.

July 11, 2024 Read

Point Power

In previous posts we talked about inscribed angle. The power of a point is a real number that reflects the relative distance of a given point to a given circle. To understand this better, imagine you have a point A outside a circle. Draw two secants of A that intersect the circle at two points each. The power of point A is then defined as the product of the distances from A to one point of intersection with the product of the distances from A to the other point of intersection on the other secant.

July 10, 2024 Read

Segment angle

In previous posts we talked about inscribed angle. The segment angle is defined as the angle formed between a chord of a circle and the tangent to the circle at one of the ends of the chord. This angle can be seen as a special case of the inscribed angle, where the vertex of the angle is precisely on one of the points that define the arc. To understand better, imagine a circle with a rope extended from one point to another. At one of the points where the rope touches the circle, draw a tangent line. The angle formed between this tangent and the chord is what we call the segment angle. An interesting property of this angle is that it is half the measure of the central angle that subtends the same arc.

July 9, 2024 Read

Inscribed angle

In previous posts we talked about properties of circular arcs. Now, we talk about angles inscribed in a circle. An inscribed angle is one whose vertex is on a circle and whose sides are secants that intersect the circle at two other distinct points. The most notable property of an inscribed angle is that its measure is equal to half the measure of the arc it subtends on the circumference.

July 4, 2024 Read

Arches Properties

In previous posts we talked about circumscribable segments. Now, we talk about properties of circular arcs. An arc of a circle is a portion of a circle bounded by two points. The circumference, in turn, is the set of all points equidistant from a central point in a plane. One of the properties is the congruence of arcs. Two arcs are congruent (that is, they have the same measure) if they subtend congruent central angles. This is useful, for example, in constructing regular polygons inscribed in a circle, where all sides and angles are equal.

July 3, 2024 Read

Circumscript segments

In previous posts we talked about positions relative to the circumference. Now, we talk about circumscriptive segments. Geometry is a fascinating area of mathematics that studies the properties and relationships of points, lines, angles, surfaces and solids. Within this vast field lie circumscriptive segments, which play a crucial role in many aspects of geometry and beyond. Circumscript segments refer to line segments that connect points on the circumference of a circle or other geometric figure. These segments are fundamental for understanding inscribed and circumscribed figures, which are figures drawn inside or outside another figure, respectively.

July 2, 2024 Read

Position relative to the Circumference

In previous posts we talked about Introduction to Circumference. Mathematics is a fascinating science that allows us to understand and describe the world around us with precision and beauty. One of the most interesting topics within geometry is the study of the relative positions between straight lines and circles. This field of study has a rich history, marked by significant discoveries made by mathematicians over the centuries. The relative positions of a straight line in relation to a circle can be classified into three main categories:

July 1, 2024 Read

Introduction to Circumference

In previous posts we talked about Polygons. Now, let’s talk about Circumference. The circle, one of the most perfect and fundamental shapes in geometry, has fascinated mathematicians throughout history. The quest to understand its nature and properties has led to significant discoveries that have shaped not only the field of mathematics but also the way we view the world. History The relationship between the diameter and length of a circle, known as π (pi), is a constant that has intrigued mathematicians for centuries. The search for this constant has led to significant advances in mathematics and understanding of the physical universe.

June 28, 2024 Read

Polygon Properties

In previous posts we talked about regular polygons. Now, let’s talk about the properties of polygons. Constituent Elements Each polygon is made up of fundamental elements such as vertices, sides, diagonals and internal and external angles. Vertices are the points where two sides meet, and sides are the line segments that connect these vertices. Diagonals are straight segments that connect two non-consecutive vertices, and internal angles are formed by the intersection of two consecutive sides.

June 28, 2024 Read

Introduction to Polygons

In previous posts we talked about triangles. Now, let’s talk about Polygons. Polygons are fascinating geometric figures that have been an integral part of mathematics and human culture since time immemorial. The word “polygon” originates from the Greek “polígonos”, which means “many angles” and reflects the nature of these shapes, limited by straight lines and a variable number of sides. History Polygons have a fascinating history, from their discovery by ancient mathematicians to their application in various areas of knowledge. The first records of polygons date back to antiquity, in the Egyptian and Mesopotamian civilizations. Ancient people already used these shapes to measure land, build monuments and even to symbolize religious concepts.

June 25, 2024 Read