Equilateral Triangle

An equilateral triangle is a triangle with the identical length of all three sides in geometry. An equilateral triangle is also equiangular in Euclidean geometry, which means that all three internal angles are congruent and each angle is 60 degrees. It is also known as a regular triangle since it is a regular polygon. Equal angles make up an equilateral triangle. As a result, all three angles satisfy the congruence criterion, making them consistent with one another. In terms of geometry, it’s also a regular polygon with three equal sides.

What is an equilateral triangle?

An equilateral triangle is a triangle with all 3 sides of identical length “a”, similar to what may also be called a “regular” triangle. An equilateral triangle is consequently a unique case of an isosceles triangle having not simply two, however all 3 sides are equal. An equilateral triangle even has 3 identical 60 degrees angles. Depending on the length of their sides, triangles are divided into three main categories. The isosceles triangle, the scalene triangle, and the equilateral triangle are the three types of triangles. The equilateral triangle differs from the isosceles and scalene triangles in that it has three sides. 

• In the scalene triangle, all of the triangle’s sides are not equal, and all of the triangle’s angles are not equal. 
• It’s an isosceles triangle if two sides are equal and the opposing angles of the same-sized sides are also equal. 
• In an equilateral triangle, all of the sides are equal, and all of the angles are equal as well. 

The circumcenter of Equilateral Triangle

The point of intersection of the perpendicular bisectors of the sides of an equilateral triangle is known as the circumcenter. In this case, the circumcircle goes across each of the triangle’s three vertices, completing the triangle. 

It is referred to be an equilateral triangle if any of the triangle’s three centers (incenter, orthocenter, and centroid) coincide with the triangle’s circumcenter. 

equilateral triangle properties

• All three sides are equal and all three angles are congruent and are equal to 60 degrees.
• The altitude, median, angle bisector, and perpendicular bisector of each side of the equilateral triangle lie on the same line.
• The three lines one for every side are also referred to as lines of symmetry.
• The equilateral triangle diagonal formula given as √3a2/4.
• The orthocenter, circumcenter, incenter, centroid are all the same point. 
• The circumradius of an equilateral triangle is a√3/3. Here, ⅔ is the length of an attitude. Because each altitude is also a triangle’s median, the length of an altitude is the same.
• The inradius of an equilateral triangle is a√3/6. Again here the inradius is ⅓ the length the altitude, because as we know altitude is also called the median of the triangle. 
• equilateral triangle diagonal formula is . ​a​2 + ​b​2 = ​c2

Theorems related to Equilateral Triangle

Experts Theorem

Morley’s Theorem:- According to Morley’s theorem, the three intersection points of adjacent angle trisectors make an equilateral triangle, 

In fact, the remaining intersection points generate four more equilateral triangles, proving the theory. This theorem can be extended to provide a total of 18 equilateral triangles.

Napoleon’s Theorem:- According to Napoleon’s theorem, if equilateral triangles are built on the sides of any triangle, the three triangles’ centres form an equilateral triangle.

The outer Napoleon triangle is formed when the triangles are erected outwards. Otherwise, the triangle is known as the inner Napoleon triangle if the triangles are erected inwards. The area of the original triangle is equal to the difference between the areas of these two triangles. The centre of the inner and outer Napoleon triangles is the same as the centroid of the original triangle.

Basic Theorems with Proofs

Theorem1: In an equilateral triangle, each angle has the same length and measures 60 degrees in width and length. 

Proof:  Assume that an equilateral triangle has the letters ABC. 

AB=AC=>C=B. — (1) due to the fact that angles opposite to equal sides are equal (This is known as the isosceles triangle theorem.) 

Also, AC=BC=>B=A   — (2) due to the fact that angles opposite to equal sides are equal (Isosceles triangle theorem) 

We have the following from (1) and (2): 

A=B=C — (3)

In △ABC,

A+B+C=180 degree (Angle sum property)

=>A+A+A=180 degree

=>A=180/3 =60 degree

Therefore, A=B=C=60 degree

Therefore the angles of the equilateral triangle are 60 degrees each.

Hence Proved.

Theorem2: The term “equilateral” refers to the property of a triangle being equal in all three directions. 

Proof: Let an equilateral triangle be ABC

AB=AC=>C=B. — (1) since angles opposite to equal sides are equal. (Isosceles triangle theorem)

Also, AC=BC=>B=A   — (2) since angles opposite to equal sides are equal. . (Isosceles triangle theorem)

Therefore, A=B=C — (3)

Hence Proved.

Equilateral Triangle Facts and Figures 

• There are 3 Sides.
• Each interior angle = 60
• Each exterior angle 120
• There are 3 angles
• Each interior angle is of 60o

Centrifugal Force of Equilateral Triangle 

The centroid of an equilateral triangle is the point in the middle of the triangle where the two sides meet. Because all of its sides are of identical length, it is simple to determine where the centroid of the triangle is. 

To locate the centroid of a triangle, we must first draw perpendiculars from each vertex of the triangle to the opposing sides of the triangle. These perpendiculars are all of identical lengths, and they all cross at a single point, which is known as the centroid of the triangle. 

Conclusion

So, here we come to an end to the knowledge and understanding of equilateral triangles. If we sum up the entire content we come to know that in equilateral triangles all sides, and angles are equal and it has different names also like a regular triangle, and equiangular triangle and to name a few. Also, we have discussed various theorems given by different experts. Considering all these facts we can say that the equilateral triangle is a special case of the isosceles triangle because it fulfills the condition of an isosceles triangle that there must be two sides equal, however, it has all sides equal.