Similarly, side BC reflects to side AB.

Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. Another useful criterion is that the three angles of an equilateral triangle are equal as well, and are thus each 60∘60^{\circ}60∘. For any point P in the plane, with distances p, q, and t from the vertices A, B, and C respectively,[19], For any point P in the plane, with distances p, q, and t from the vertices, [20]. a+bω+cω2=0,a+b\omega+c\omega^2 = 0,a+bω+cω2=0, If the triangles are erected outwards, as in the image on the left, the triangle is known as the outer Napoleon triangle.

2

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). Suppose, ABC is an equilateral triangle, then, as per the definition; AB = BC = AC, where AB, BC and AC are the sides of the equilateral triangle.

Log in here. [16]:Theorem 4.1, The ratio of the area to the square of the perimeter of an equilateral triangle, Because the equilateral triangle is, in some sense, the simplest polygon, many typically important properties are easily calculable.

4 Properties of an equilateral triangle.A triangle with three equal sides is equilateral. Log in. If P is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem.

In an isosceles triangle, the base angles are congruent. The three altitudes of an equilateral triangle intersect at a single point. 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.

Includes formulas, pictures, of equilateral triangles

For other uses, see, Six triangles formed by partitioning by the medians, Chakerian, G. D. "A Distorted View of Geometry."

[14]:p.198, The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral. where R is the circumscribed radius and L is the distance between point P and the centroid of the equilateral triangle.

The difference between the areas of these two triangles is equal to the area of the original triangle. A There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral.

The integer-sided equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees.

However, the first (as shown) is by far the most important. Find p+q+r.p+q+r.p+q+r.

In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.

{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} Altitudes of equilateral triangles.

By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2.

An equilateral triangle is easily constructed using a straightedge and compass, because 3 is a Fermat prime. find the measure of ∠BPC\angle BPC∠BPC in degrees. Firstly, it is worth noting that the circumradius is exactly twice the inradius, which is important as R≥2rR \geq 2rR≥2r according to Euler's inequality. In flat Euclidean geometry are those triangles that have all their sides equal.

Three of the five Platonic solids are composed of equilateral triangles. Altitude CD divides equilateral triangle △ABC into two 30°-60°-90° triangles. The geometric center of the triangle is the center of the circumscribed and inscribed circles, The height of the center from each side, or, The radius of the circle circumscribing the three vertices is, A triangle is equilateral if any two of the, It is also equilateral if its circumcenter coincides with the.

The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. 1 □MA=MB+MC.\ _\squareMA=MB+MC.

t An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center.

An alternative method is to draw a circle with radius r, place the point of the compass on the circle and draw another circle with the same radius.

if t ≠ q; and. 3

The sides of rectangle ABCDABCDABCD have lengths 101010 and 111111. of 1 the triangle is equilateral if and only if[17]:Lemma 2.

A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid. A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. In both methods a by-product is the formation of vesica piscis.

□​.

Since an equilateral triangle is also an equiangular triangle, it is a regular polygon. Triangles △ACD and △BCD both have legs of length , and hypotenuse s. △ABC is an equilateral triangle since AB≅AC≅BC.

= Isosceles Triangle Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is the incircle). An equilateral triangle is also called an equiangular triangle since its three angles are equal to 60°.

{\displaystyle \omega } [22], The equilateral triangle is the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes) (the heptagonal triangle being the only obtuse one).[23]:p.

An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.

Its symmetry group is the dihedral group of order 6 D3. An isosceles triangle has at least two equal sides, so an equilateral triangle is also an isosceles triangle.

Equilateral triangle .

The most straightforward way to identify an equilateral triangle is by comparing the side lengths.

https://brilliant.org/wiki/properties-of-equilateral-triangles/. Given that △ABC\triangle ABC△ABC is an equilateral triangle, with a point PP P inside of it such that. Show that there is no equilateral triangle in the plane whose vertices have integer coordinates.

[18] This is the Erdős–Mordell inequality; a stronger variant of it is Barrow's inequality, which replaces the perpendicular distances to the sides with the distances from P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross the sides (A, B, and C being the vertices).

12 Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles:[8].

It is also a regular polygon, so it is also referred to as a regular triangle.

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. That is, PA, PB, and PC satisfy the triangle inequality that the sum of any two of them is greater than the third. And ∠A = ∠B = ∠C = 60° Based on sides there are other two types of triangles: 1. Here is an example related to coordinate plane.

The three altitudes extending from the vertices A, B, and C of △ABC above intersect at point G. Since the altitudes are the angle bisectors, medians, and perpendicular bisectors, point G is the orthocenter, incenter, centroid, and circumcenter of the triangle.