Obviously, an isosceles triangle, as well as other types of elementary triangles, belongs to the simplest and most comprehensible geometric figures, which are studied by pupils during first math lessons in school. However, the more surprising fact that many students do not pay attention to the properties of this figure, which doubtlessly leads to frustrating consequences when they are faced with increasingly complex figures. Therefore, it is quite recommended to refresh one’s memory about all properties that are characteristic for an isosceles triangle in order to gain confidence that she is ready to continue her math educational course. In fact, the main objective of this article is to remind students about different obvious properties of this elementary geometric figure and supply them with significant information about different theorems that are connected with the object of study. Thereby, let us start with the definition of an isosceles triangle.
In math, an isosceles triangle is defined as a triangle, which has two sides of equal length. Nevertheless, this definition may significantly differ in details according to the specific issues. For example, some authors define it as a triangle that has two and only two sides of equal length. In addition, an equilateral triangle can be also regarded as a special case of an isosceles triangle because it has three sides of equal length. In fact, Euclid defined this triangle as one, which has exactly two equal sides. However, in modern scientific literature, the definition of this triangle as one that has at least two equal sides is conventional. In order to simplify one’s visualization of this geometric figure, it is usually described as a triangle, which rests on a third side, directing the two equal sides upwards. Thereby, in accordance with this visualization, the third side is called the base, whereas the equal sides are called the legs of a triangle. Due to this fact, it is easy to understand why the base angles are the angles that have the base as one of their sides and the vertex angle is the angle formed by the legs of the triangle.
In fact, this type of elementary triangles has geometric properties that are characteristic for triangles in general, including a few unique peculiarities. It has only one axis of symmetry that passes through the midpoint of the base and the vertex angle. Therefore, it is obvious that the axis of symmetry in this triangle coincides with the median drawn to the base and the perpendicular bisector of the base, as well as with the angle bisector of the vertex angle and the altitude drawn from the vertex angle.
An isosceles triangle can be defined as a right, acute or obtuse triangle according to the characteristics of the vertex angle. Obviously, in classical Euclidean geometry, the base angles cannot be obtuse or right due to the fact that in this case their measures would sum to at least 180°, which is the total of all angles in any Euclidean triangle. Therefore, the type of a triangle depends only on the properties of the vertex angle. If the vertex angle is obtuse (greater than 90°), the triangle is also obtuse, if it is right (equal to 90°) than the triangle is right and if it is acute - the triangle is acute. The isosceles triangle that includes one right angle (vertex angle) is called a ‘right isosceles triangle’.
In fact, one of the characteristic properties of an isosceles triangle is that its axis of symmetry coincides with the Euler line of a triangle. The Euler line is a central line of a triangle, which intersects a set of significant points of any triangle, including the centroid, the Exeter point, the circumcenter, the orthocenter and the center of the nine-point circle of a triangle. In our case, the most interesting are the orthocenter of a triangle that is the intersection of three altitudes of a triangle, the triangle’s circumcenter, which is defined as the point of the intersection of its three sides’ perpendicular bisectors, and the triangle’s centroid (the specific point in which intersect three medians of a triangle). Therefore, we can state that the Euler line of an isosceles triangle coincides not only with its axis of symmetry but also with its perpendicular bisector and median that pass through the vertex angle. Moreover, this fact leads us to the conclusion that the location of the orthocenter, the centroid and the circumcenter of an isosceles triangle depends on its type. If the triangle’s vertex angle is acute (and so is the triangle itself, according to the previously listed statements) then these points are located inside the triangle. If the triangle is obtuse then the triangle’s circumcenter lies outside it, whereas the triangle’s centroid is located inside the triangle. In addition, it has to be noted that the incenter of the isosceles triangle is located on the Euler line.
Among various theorems that have a direct connection with this topic, the most essential for students is the theorem that characterizes the main property of an isosceles triangle: the ratio of its sides, It exists in two different versions, which determine a triangle using its sides or angles. The first version postulates that if two sides of a triangle are congruent, then the angles that are opposite of them are also congruent. Eventually, the converse version of this theorem postulates that if two angles of a triangle are congruent than two opposite triangle’s sides are also congruent according to each other.
In truth, an isosceles triangle is one of the best training models for pupils. As we already know a perpendicular bisector of the base in an isosceles triangle coincides with its axis of symmetry. Therefore, a perpendicular bisector of the base forms two congruent right triangles. One can easily prove this statement by examining any sample of standard school mathematical textbooks. Using the Pythagoras’ theorem in order to find sides of these triangles, we can solve our isosceles triangle. In order to consolidate one’s knowledge about this useful property of an isosceles triangle let us examine a few practical assignments that require profound comprehension of the essential principles that are the basis of Euclidian geometry. Here is a concise list of these assignments together with a brief explanation to each of them.
Using these examples, one can easily accomplish a lion’s share of mathematical assignments that are related to the topic of an isosceles triangle. Additionally, one can use these examples of solutions in various mathematical tasks that are more sophisticated. In fact, all geometry is based on the principles of the movement from simple to complex conceptions. Thereby, all the knowledge obtained during the study of different groups of isosceles triangles will inevitably find their application for diverse types of mathematical objectives.