Properties of Equilateral Triangles
An equilateral triangle is a triangle whose three sides all have the same length. They are the only regular polygon with three sides, and appear in a variety of contexts, in both basic geometry and more advanced topics such as complex number geometry and geometric inequalities.\[\]
Contents
Identification
The most straightforward way to identify an equilateral triangle is by comparing the side lengths. If the three side lengths are equal, the structure of the triangle is determined (a consequence of SSS congruence). However, this is not always possible.
Another useful criterion is that the three angles of an equilateral triangle are equal as well, and are thus each \(60^{\circ}\). Since the angles opposite equal sides are themselves equal, this means discovering two equal sides and any \(60^{\circ}\) angle is sufficient to conclude the triangle is equilateral, as is discovering two equal angles of \(60^{\circ}\).
Notably, the equilateral triangle is the unique polygon for which the knowledge of only one side length allows one to determine the full structure of the polygon. For example, there are infinitely many quadrilaterals with equal side lengths (rhombus) so you need to know at least one more property to determine its full structure. In this way, the equilateral triangle is in company with the circle and the sphere whose full structures are determined by supplying only the radius.
Basic Properties
Because the equilateral triangle is, in some sense, the simplest polygon, many typically important properties are easily calculable. For instance, for an equilateral triangle with side length \(\color{red}{s}\), we have the following:
- The altitude, median, angle bisector, and perpendicular bisector for each side are all the same single line.
- These 3 lines (one for each side) are also the lines of symmetry of the triangle.
- All three of the lines mentioned above have the same length of \(\frac{s\sqrt{3}}{2}\).
- The area of an equilateral triangle is \(\frac{s^2\sqrt{3}}{4}\).
- The orthocenter, circumcenter, incenter, centroid and nine-point center are all the same point. The Euler line degenerates into a single point.
- The circumradius of an equilateral triangle is \(\frac{s\sqrt{3}}{3}\). Note that this is \(\frac{2}{3}\) the length of an altitude, because each altitude is also a median of the triangle.
- The inradius of an equilateral triangle is \(\frac{s\sqrt{3}}{6}\). Note that the inradius is \(\frac{1}{3}\) the length of an altitude, because each altitude is also a median of the triangle. Also the inradius is \(\frac{1}{2}\) the length of a circumradius.
It is also worth noting that six congruent equilateral triangles can be arranged to form a regular hexagon, making several properties of regular hexagons easily discoverable as well. For example, the area of a regular hexagon with side length \(s\) is simply \(6 \cdot \frac{s^2\sqrt{3}}{4}=\frac{3s^2\sqrt{3}}{2}\).
Advanced Properties
Firstly, it is worth noting that the circumradius is exactly twice the inradius, which is important as \(R \geq 2r\) according to Euler's inequality. The equilateral triangle provides the equality case, as it does in more advanced cases such as the Erdos-Mordell inequality.
If \(P\) is any point inside an equilateral triangle, the sum of its distances from three sides is equal to the length of an altitude of the triangle:
The sum of the three colored lengths is the length of an altitude, regardless of P's position
The equilateral triangle is also the only triangle that can have both rational side lengths and angles (when measured in degrees).
When inscribed in a unit square, the maximal possible area of an equilateral triangle is \(2\sqrt{3}-3\), occurring when the triangle is oriented at a \(15^{\circ}\) angle and has sides of length \(\sqrt{6}-\sqrt{2}:\)
Both blue angles have measure \(15^{\circ}\)
It is also worth noting that besides the equilateral triangle in the above picture, there are three other triangles with areas \(X, Y\), and \(Z\) \((\)with \(Z\) the largest\().\) They satisfy the relation \(2X=2Y=Z \implies X+Y=Z \). In fact, \(X+Y=Z\) is true of any rectangle circumscribed about an equilateral triangle, regardless of orientation.
The sides of rectangle \(ABCD\) have lengths \(10\) and \(11\). An equilateral triangle is drawn so that no point of the triangle lies outside \(ABCD\). The maximum possible area of such a triangle can be written in the form \(p\sqrt{q}-r\), where \(p, q,\) and \(r\) are positive integers, and \(q\) is not divisible by the square of any prime number. Find \(p+q+r.\)
Equilateral triangles are particularly useful in the complex plane, as their vertices \(a,b,c\) satisfy the relation \[a+b\omega+c\omega^2 = 0,\] where \(\omega\) is a primitive third root of unity, meaning \(\omega^3=1\) and \(\omega \neq 1\). In particular, this allows for an easy way to determine the location of the final vertex, given the locations of the remaining two.
Another property of the equilateral triangle is Van Schooten's theorem:
If \(ABC\) is an equilateral triangle and \(M\) is a point on the arc \(BC\) of the circumcircle of the triangle \(ABC,\) then
\[MA=MB+MC.\]
Using the Ptolemy's theorem on the cyclic quadrilateral \(ABMC\), we have
\[MA\cdot BC= MB\cdot AC+MC\cdot AB\]
or
\[MA=MB+MC.\ _\square\]
Here is an example related to coordinate plane.
Show that there is no equilateral triangle in the plane whose vertices have integer coordinates.
Suppose that there is an equilateral triangle in the plane whose vertices have integer coordinates.
The determinant formula for area is rational, so if the all three points are rational points, then the area of the triangle is also rational.
On the other hand, the area of an equilateral triangle with side length \(a\) is \(\dfrac{a^2\sqrt3}{4}\), which is irrational since \(a^2\) is an integer and \(\sqrt{3}\) is an irrational number.
This is a contradiction. \(_\square\)
Theorems involving Equilateral Triangles
Morley's theorem states that the three intersection points of adjacent angle trisectors form an equilateral triangle (the pink triangle in the picture on the right).
In fact, this theorem generalizes: the remaining intersection points determine another four equilateral triangles. Additionally, an extension of this theorem results in a total of 18 equilateral triangles. However, the first (as shown) is by far the most important.
Napoleon's theorem states that if equilateral triangles are erected on the sides of any triangle, the centers of those three triangles themselves form an equilateral triangle.
If the triangles are erected outwards, as in the image on the left, the triangle is known as the outer Napoleon triangle. Otherwise, if the triangles are erected inwards, the triangle is known as the inner Napoleon triangle. The difference between the areas of these two triangles is equal to the area of the original triangle.
The inner and outer Napoleon triangles share the same center, which is also the centroid of the original triangle.
