Introduction

The Importance of Triangles, Cevians, and Notable Points

Triangles are elementary geometric figures, present in a wide variety of phenomena and structures in our daily lives. They play a crucial role in Mathematics, providing a foundation for many concepts and theorems.

Cevians, lines that connect the vertices of a triangle to the midpoints of the opposite sides, not only contribute to the beauty and symmetry of the triangle but are especially interesting because they intersect at a point called the centroid or center of gravity of the triangle. This point is a special case of a 'notable point,' a point that has unique properties in relation to the segments and angles of the triangle.

The notable points of a triangle - centroid, incenter, circumcenter, and orthocenter - are incredibly rich and varied. They have mathematical properties that may seem surprising at first, but when explored and understood, lead to a greater appreciation and understanding of geometry and its applications.

Therefore, in this Lecture Note, we will delve into the fascinating intersection of triangles, cevians, and notable points, exploring their definitions, properties, and demonstrations.

Let's get started!

Theoretical Development

Components

• Triangles: Triangles are flat figures that have three sides, three internal angles, and three vertices. Their fundamental property is that the sum of the internal angles always totals 180 degrees, which is known as the 'sum of the interior angles of a triangle.' They vary in terms of side lengths and angle sizes, giving rise to different types of triangles (equilateral, isosceles, scalene) with different properties.

• Cevians: Cevians are the lines that connect each vertex of a triangle to the midpoint of the opposite side. Each triangle has exactly three cevians. Cevians play a fundamental role in defining notable points, such as the centroid.

• Notable Points: Notable points of a triangle are special points that have unique properties in relation to the segments and angles of the triangle. There are four notable points in each triangle: centroid, incenter, circumcenter, and orthocenter. Each of these notable points can be defined in terms of cevians. For example, the centroid is the point of intersection of the cevians, while the incenter is the center of the circle inscribed in the triangle.

Key Terms

• Centroid: The centroid of the triangle, also known as the center of gravity, is the point of intersection of the cevians. For a triangle with vertices A, B, and C, the centroid is denoted as G. It fulfills the property of dividing each of the cevians in the ratio 2:1.

• Incenter: The incenter of the triangle is the center of the circle inscribed in the triangle. Denoted by I, it is equidistant from the three sides of the triangle.

• Circumcenter: The circumcenter of the triangle is the center of the circle circumscribed around the triangle. Denoted by O, it is equidistant from the three vertices of the triangle.

• Orthocenter: The orthocenter is the point of intersection of the altitudes of the triangle. Denoted by H, it is the only point among the four notable points that can be outside the boundaries of the triangle.

Examples and Cases

• Euler's Theorem: An interesting case of interaction between notable points is Euler's Theorem. This theorem states that, in any triangle, the distance between the centroid (G), the circumcenter (O), and the incenter (I) satisfies the relation OG = 2GI. Furthermore, OH = 3GI.

• Calculation of the Centroid: For a triangle ABC, with the coordinates of the vertices A(x1, y1), B(x2, y2), and C(x3, y3), the centroid G has the following coordinates: G[ (x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3].

• Calculation of the Incenter: For a triangle ABC, if the coordinates of the vertices are A(x1, y1), B(x2, y2), and C(x3, y3), and the lengths of the sides opposite the vertices are a, b, and c respectively, then the incenter I has the following coordinates: I[ (ax1 + bx2 + cx3) / (a + b + c), (ay1 + by2 + cy3) / (a + b + c)].

Detailed Summary

Key Points

• Triangle Definition: It is a geometric shape that has three sides, three internal angles, and three vertices. The sum of the internal angles of a triangle is always 180 degrees.

• Importance of Cevians: They are lines that connect each vertex of a triangle to a midpoint on the opposite side. Cevians are key to defining notable points.

• Notable Points in a Triangle: There are four notable points in each triangle: centroid, incenter, circumcenter, and orthocenter. These points have unique properties that are intrinsically linked to cevians.

• Centroid Point: It is the center of gravity of the triangle, defined as the intersection of the cevians. It divides each cevian in the ratio 2:1.

• Incenter Point: It is the center of the circle inscribed in the triangle. It is equidistant from the three sides of the triangle.

• Circumcenter Point: It is the center of the circle that circumscribes the triangle. It is equidistant from each vertex of the triangle.

• Orthocenter Point: It results from the intersection of the altitudes of the triangle and is the only point among the four notable points that can be outside the boundaries of the triangle.

• Euler's Theorem: It relates the distance between the centroid, circumcenter, and incenter in a triangle. It states that the distance from the centroid to the circumcenter is twice the distance from the centroid to the incenter. Additionally, the distance from the orthocenter to the centroid is three times the distance from the incenter to the centroid.

• Calculation of Notable Points: It is possible to calculate the coordinates of the notable points of a triangle, demonstrating the mathematical concreteness of these points and their relationships with the vertices of the triangle.

Conclusions

• Abundant Understanding of Triangles: After exploring the topics of cevians and notable points, we have had the opportunity to learn more about triangles, one of the most basic and important figures in Geometry.

• Interconnection between Components: The notable relationship between the components - triangles, cevians, and notable points - is evidence of the beauty and coherence of Mathematics. Each component influences and is connected to the others in intriguing ways.

• Applications Beyond Theory: The properties studied, such as Euler's Theorem and the equations to calculate the notable points, can be applied to solve practical problems in Mathematics, Physics, Engineering, Architecture, and many other disciplines.

Exercises

1. Recognizing Notable Points: Given a triangle ABC, use the definition of notable points to locate each of them.

2. Properties of Notable Points: Demonstrate that the centroid divides each cevian in the ratio 2:1. Also, prove that the incenter is equidistant from the three sides of the triangle.

3. Euler's Theorem: Verify if Euler's Theorem is true for a specific triangle. Calculate the distances between its centroid, circumcenter, and incenter.