# What is an isosceles triangle: a definition of the term

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.

## The main geometric properties of an isosceles 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.

### How to solve an isosceles triangle: a brief practical guideline

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.

• Finding the base of the triangle. In order to find the base given the leg and altitude, one should use the formula: the base = 2(L2 – A2)1/2. In this formula: L is the length of the leg and A is the altitude.
• Finding the leg of the triangle. In order to find the leg length given the base and altitude, one should use the formula: the leg = (A2 + (B/2)2)1/2. In this formula: B is the length of the base and A is the altitude.
• Finding the altitude of the triangle. In order to find the altitude given the base and leg, one should use the formula: the altitude = (L2 – (B/2)2)1/2. In this formula: L is the length of the leg and B is the base.

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.

# Our Service Charter

1. ### Excellent Quality / 100% Plagiarism-Free

We employ a number of measures to ensure top quality essays. The papers go through a system of quality control prior to delivery. We run plagiarism checks on each paper to ensure that they will be 100% plagiarism-free. So, only clean copies hit customers’ emails. We also never resell the papers completed by our writers. So, once it is checked using a plagiarism checker, the paper will be unique. Speaking of the academic writing standards, we will stick to the assignment brief given by the customer and assign the perfect writer. By saying “the perfect writer” we mean the one having an academic degree in the customer’s study field and positive feedback from other customers.
2. ### Free Revisions

We keep the quality bar of all papers high. But in case you need some extra brilliance to the paper, here’s what to do. First of all, you can choose a top writer. It means that we will assign an expert with a degree in your subject. And secondly, you can rely on our editing services. Our editors will revise your papers, checking whether or not they comply with high standards of academic writing. In addition, editing entails adjusting content if it’s off the topic, adding more sources, refining the language style, and making sure the referencing style is followed.
3. ### Confidentiality / 100% No Disclosure

We make sure that clients’ personal data remains confidential and is not exploited for any purposes beyond those related to our services. We only ask you to provide us with the information that is required to produce the paper according to your writing needs. Please note that the payment info is protected as well. Feel free to refer to the support team for more information about our payment methods. The fact that you used our service is kept secret due to the advanced security standards. So, you can be sure that no one will find out that you got a paper from our writing service.
4. ### Money Back Guarantee

If the writer doesn’t address all the questions on your assignment brief or the delivered paper appears to be off the topic, you can ask for a refund. Or, if it is applicable, you can opt in for free revision within 14-30 days, depending on your paper’s length. The revision or refund request should be sent within 14 days after delivery. The customer gets 100% money-back in case they haven't downloaded the paper. All approved refunds will be returned to the customer’s credit card or Bonus Balance in a form of store credit. Take a note that we will send an extra compensation if the customers goes with a store credit.

We have a support team working 24/7 ready to give your issue concerning the order their immediate attention. If you have any questions about the ordering process, communication with the writer, payment options, feel free to join live chat. Be sure to get a fast response. They can also give you the exact price quote, taking into account the timing, desired academic level of the paper, and the number of pages.

Excellent Quality
Zero Plagiarism
Expert Writers

or

Instant Quote