Using GeoGebra in generalization processes of geometrical challenging problems1

We generalize in this work three geometrical challenging problems addressed in mathematics literature. In generalizations, we adopt the theoretical assumptions established for this process and use GeoGebra to build figures and animation. The proposed and solved generalizations establish natural links between some mathematics areas, highlighting the importance of generalization processes for constructing mathematical knowledge in undergraduate programs in mathematics teacher education. We conclude that the use of GeoGebra was essential to a comprehensive understanding of the structures for generalization.


Introduction
considers that every concept is the result of a generalization process. Thus, for him, all concepts learned by human beings, and here we highlight the concepts and mathematical properties, are internalized through a generalization process.
According to Dumitrascu (2017, p. 47): In the mathematics literature, generalization can be seen as a statement that is true for a whole category of objects; it can be understood as the process through which we obtain a general statement; or it can be the way to transfer knowledge from one setting to a different one.
In accordance with Hashemi et al. (2013), the generalization is one of the fundamental activities in learning mathematics, which needs to be further explored by individuals who teach and study mathematics. For Mason (1996), generalization is the heartbeat of mathematics. Davydov (1990) argues that the development of student generalization capacity is one of the main goals of mathematics, while Sriraman (2004) considers that the generalization begins with the construction of examples, within which plausible patterns are detected and lead to the formulation of theorems.
However, generalizing in mathematics, particularly in geometry (ALLEN, 1950;PARK;KIM, 2017), is generally not a trivial process. From the sum of the internal angles of a triangle (180°) to the sum of the internal angles of an n-sided convex polygon (180°( − 2)), the generalization occurs by a partition of thesided convex polygon into − 2 triangles; from the Pythagorean theorem ( 2 + 2 = 2 ) to the Fermat theorem ( + = ), the generalization-required centuries of study and the creation of new areas in mathematics (SINGH, 2002).
In this way, we illustrate in this work the generalization process in geometry using the concepts of Sriraman (2004). In this process, we replaced the construction of examples by selecting three geometrical challenging problems from the book Challenging problems in geometry by Posamentier and Salkind (1996): the measure of the midsegment of a triangle, the section of the hypotenuse, and the section of an internal angle of a triangle. These three geometrical challenging problems are then transformed into theorems, which can be complemented with proofs without words (NELSEN, 1993;LAGO;NÓS, 2020;FERNANDES, 2018FERNANDES, , 2019 in the dynamic geometry software GeoGebra (2021) and can be approached in mathematics teacher training courses. The first of the three problems can be presented in high school math classes.

Midsegment of a triangle
Problem 1 (Challenging problem 3-7, page 12) If the measures of two sides and the included angle of a triangle are 7, √50, and 135°, respectively, find the measure of the segment joining the midpoints of the two given sides.
Solution 1 Consider the triangle , with sides = = 7, = = √50 and = , angle ̂= 135°, E, and F midpoints on sides ���� and ���� , respectively, and point D, orthogonal projection of the vertex of the triangle on the extension of side ���� , as shown in Figure 1. Applying the Pythagorean theorem (LOOMIS, 1968) in the isosceles right triangle , with measurement sides = = , we obtain that.
⁂ By fixing the measurements of the two sides of a triangle, as in Problem 1, and varying the measurement of the angle determined by those sides, we can visually check or make a proof without words of the triangle midsegment theorem using a dynamic geometry software. We built an animation in GeoGebra, which can be done in mathematics classroom, and we make it available at the following address: https://www.GeoGebra.org/m/hamgfbgj. This animation makes it easier to devise the generalization of Problem 1.
Theorem 1 (Generalization of Problem 1) If and are the measures of two sides of a triangle and is the angle determined by these two sides, then the measure of the segment whose ends are the midpoints of the sides with measures and is equal to Proof Considers in the triangle , with sides = , = , and = : the angle ̂= , 90°< < 180°; points E and F, respectively, midpoints of the sides ���� and ���� , and point , orthogonal projection of the vertex of the triangle on the extension of side ���� , as shown in Figure 2. Calculating trigonometric ratios in the right triangle , where = and = , and determining trigonometric transformations (HILL, 2019), we have (3) Applying the Pythagorean theorem in the right triangle - Figure 2, and using (2), and (3), and a trigonometric identity (HILL, 2019), we conclude that where is the measure of the hypotenuse of the right triangle .
Since is ���� a midsegment of the triangle , we have by the triangle midsegment theorem that We can show that relations (4) and (5) remain true if 0°< ≤ 90°.

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The relation (4) is the law of cosines (HILL, 2019), and it can be applied directly to the triangle - Figure 1, to solve Problem 1. However, we choose to deduce it through the generalization of Problem 1, thus showing that generalization processes can be used in mathematics classes as demonstration activities.
Using in relation (5) standard values of the first and second quadrants for the angle , we have In relation with = 30°, 45°, 135°, 150°, we have nested radicals, which allow us to discuss in the classroom the rules for denesting radicals (GKIOULEKAS, 2017; NÓS; SAITO; SANTOS, 2017).

Section of the hypotenuse
Problem 2 (Challenging problem 10-4, page 46) Prove that the sum of the squares of the distances from the vertex of the right angle, in a right triangle, to the trisection points along the hypotenuse, is equal to 5 9 ⁄ the square of the measure of the hypotenuse. Posamentier and Salkind (1996) propose using Stewart's theorem (page 45) to solve challenging problem 2. They also propose in challenge 2 of problem 10-4 to predict the value of the sum of the squares for a quadrisection of the hypotenuse (NÓS; SAITO; OLIVEIRA, 2016).
⁂ 5 Cevian is a line segment that joins a vertex of a triangle with a point on the opposite side (or its extension).
In this way of thinking, calculating the sum of the squares of the measures of the cevians that section the hypotenuse in congruent ( = 2,3,4, … ) segments, we will find, respectively, the following fractions of the square of the measure of the hypotenuse: So, the question to be answered is whether we can establish the nth term of the sequence (8).
Theorem 2 (Generalization of Problem 2) If , = 1,2, … , , are the measurements of the cevians with an end at the vertex of the right angle of a right triangle and which divide the hypotenuse in congruent + 1 segments, then where is the measure of the hypotenuse of the right triangle.
□ Equation (9) can be explored in GeoGebra. We build an animation for = 1,2,3, and we make it available at

Section of an internal angle of a triangle
Problem 3 (Challenging problem 10-6, page 46) Prove that in any triangle the square of the measure of the internal bisector of any angle is equal to the product of the measures of the sides forming the bisected angle decreased by the product of the measures of the segments of the side to which this bisector is drawn. Posamentier and Salkind (1996) propose two strategies to solve challenging problem 3: use Stewart's theorem (page 45) and, subsequently, the internal bisector theorem (theorem 47, page 242); or use properties of the inscribed quadrilateral (theorems 36a and 37, page 242) to establish similar triangles and, soon after, the intersecting chords theorem (theorem 52, page 243). The second strategy minimizes algebraic work in the generalization of challenging problem 3.

Solution 3 Considers: the triangle
, with sides = , = , and = ; the segment ���� with ∈ ���� of measure 1 , which divides the angle ̂ in two congruent angles of measure , and divides the side ���� in the segments = 1 and = 2 ; the point 1 , belonging to the extension of the segment ���� and the circumference that circumscribes the triangle , as shown in Figure 5.
Applying the intersection chords theorem, we obtain Replacing (16) in (15), we conclude that

⁂
To generalize challenging problem 3 we should detect patterns (SRIRAMAN, 2004). Let us begin by analyzing two particular cases: the square of the measure of the segments that divide an internal angle of a triangle in three and four congruent angles.
For the cevians 1 and 2 that divide the angle in three congruent angles, as illustrated in Figure 6(a), we conclude, using the same strategy, that (a) (b) FIGURE 6: Generalization of challenging problem 3: (a) cevians that divide angle ̂ in three congruent angles; (b) cevians that divide angle ̂ in four congruent angles SOURCE: Authors with GeoGebra Now, for the cevians 1 , 2 and 3 that divide the angle in four congruent angles, as shown in Figure 6(b), we find that After/before proposing the generalization of Problem 3, we can observe the partition of an internal angle of a triangle in GeoGebra. We build an animation for two cases: bisection and trisection of the angle ̂, and we make it available at https://www.GeoGebra.org/m/fhhgsfja.
Equations (17)-(21) show that, in the generalization of challenging problem 3, the square of the measure of the cevians cannot be expressed, except in particular cases as in equation (20), depending only on the measurements of the sides that determine the sectioned angle and of the segments determined by the cevians on the side opposite to the sectioned angle. When calculating the measures , = 1,2, … , , it is necessary to solve a system of non-linear equations for ≥ 2.
Theorem 3 (Generalization of Problem 3) If , = 1,2, … , , are the measures of the cevians that divide an internal angle of a triangle in congruent + 1 angles, then where and are the measurements of the sides that determine the sectioned angle, and 1 , 2 , … , +1 are the measurements of the segments determined by the cevians on the side opposite to the sectioned angle.
For any , = 1,2, … , , we have by the property of the inscribed quadrilateral that � ≡̂= , as shown in Figure 7

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In the generalization of challenging problem 3, we found that, given the measures of the sides = and = , which determine the sectioned angle of the triangle , and the measures 1 , 2 , … , , +1 of the segments determined by the cevians on the opposite side ���� to the sectioned angle, it is possible to calculate the measurements of cevians by solving the following system of nonlinear equations: The system of non-linear equations (24) is a decoupled system with two equations, because if is odd, we get +1 2 Additionally, then we have an even number of equations. Thus, to solve the system (24), it is sufficient to solve a system of non-linear equations with two equations and two unknowns, i.e. fixing , = 1,2, … , , solve the following nonlinear equation system: . (25) In Proposition 1, we prove that the system of non-linear equations (25) has a solution.
Since and are positive numbers, we conclude that Finally, replacing (29) in the second equation of the system (27), we obtain 2 = � − .
Thus showing that the system (25) has a solution since > � .

Concluding remarks
In this work we present the generalization of three geometrical challenging problems proposed in Challenging problems in geometry by Posamentier and Salkind (1996). In generalization procedures, we establish a link between some areas of mathematics, such as geometry and arithmetic, we involve some theorems of plane geometry, and we use the software GeoGebra to construct figures and animation.
It is important to emphasize that, as challenging problem 3 shows, the generalization process can lead to changes in the configuration of the expected result, that is, the theorem thesis proposed in cited case.
Following Hashemi et al. (2013), we hope that this work will motivate the main agents involved in the mathematics teaching-learning process, i.e. students and teachers, to establish generalization processes in the classroom in the analysis/investigation of mathematical properties, particularly in geometry, thus contributing to consolidation and expansion of mathematical knowledge.