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Equilateral Triangle Motivation: Equilateral Triangle is a term we use for the triangles with equal length for three sides. After studying so many theorems for geometry, a lot of good properties of equilateral triangle can be derived quite easily. We just use this topic to refresh the theorems we learned before.
Theorem ( Angle of Equilateral Triangle ) Each interior angle of equilateral triangle is 600 .
Proof: ABC is an equilateral triangle. In other words, = = . Let’s start the proof below.
1. = B = C ( property of isosceles triangle )
Similarly, = A = C
Thus, A= B= C
2. The sum of interior angles of a triangle is 1800. A= B= C = 600
Theorem ( Center of Equilateral Triangle ) The incenter, circumcenter , center of inscribed circle, centroid, and orthocenter of an equilateral triangle is the same point.
Proof: 1. In an equilateral triangle ABC, find the midpoint of and denote this midpoint as M. is median from the side . 2. From the property of isosceles triangle, . In other words, is the perpendicular bisector of ; is also the altitude from .
3. Furthermore, BAM = CAM . is also the bisector of BAC .
So, is altitude, perpendicular bisector, bisector of an interior angle, median, and altitude. And ABC is an equilateral triangle. Thus, every side is with the same property like this. The 5 “centers” are the same point.
Theorem ( Altitude of Equilateral Triangle ) ABC is an equilateral triangle. . Then
=
Proof: 1. = , From the property of isosceles triangle, we know that = ( from each side, the altitude is unique. You can find other point on with the same property ). ( Please note that you can not use SSA to claim this point. There is no SSA congruence theorem. But if you can use RHS theorem for the right angle triangle. )
2. = = = 2 , and from Pythagorean Theorem 2 = 2 - 2 = 2 So, = =
Theorem ( Radius of Inscribed Circle of Equilateral Triangle ) ABC is an equilateral triangle. O is the center of the inscribed circle of ABC. Then the radius of the inscribed circle is
r =
Proof: 1. ABC is an equilateral triangle Incenter = Centroid = Orthocenter=circumcenter Thus, we can just use the property of centroid . = 2
2. From previous theorem, we have = = = = =
The radius of inscribed circle is because is the angle bisector and .
Corollary ( Radius of Circum-circle of Equilateral Triangle ) The circle passing A, B, C is known as circum-circle. Its radius is .
Theorem ( Area of Equilateral Triangle ) ABC is an equilateral triangle. Then ABC = 2
This theorem can be easily derived from the result of previous theorem. |
