Title: Equilateral triangle
Content:  In geometry, an equilateral triangle is a triangle in which all three sides have the same length. 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°. It is also a regular polygon, so it is also referred to as a regular triangle.
Title: Equilateral triangle
Content: The three altitudes have equal lengths. ; The three medians have equal lengths. ; The three angle bisectors have equal lengths. Three kinds of cevians coincide, and are equal, for (and only for) equilateral triangles:
Title: Equilateral triangle
Content:  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. 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. The ratio of the area of the incircle to the area of an equilateral triangle, which is the optic equation. There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral. An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the dihedral group of order 6 D3. Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is 
Title: Configuration (polytope)
Content:  There are three symmetry classifications of a triangle: equilateral, isosceles, and scalene. They all have the same incidence matrix, but symmetry allows vertices and edges to be collected together and counted. These triangles have vertices labeled A,B,C, and edges a,b,c, while vertices and edges that can be mapped onto each other by a symmetry operation are labeled identically.
Title: Equilateral triangle
Content:  [[File:Equilateral-triangle-heights.svg|300px|thumb|An equilateral triangle. It has equal sides ( In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors, and the medians to each side coincide.
Title: Equilateral triangle
Content: A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius. ; A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid. For any triangle, the three medians partition the triangle into six smaller triangles.
Title: Equilateral triangle
Content: A triangle is equilateral if and only if, for every point P in the plane, with distances p, q, and r to the triangle's sides and distances x, y, and z to its vertices, 
Title: Equilateral triangle
Content: A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide. ; It is also equilateral if its circumcenter coincides with the Nagel point, or if its incenter coincides with its nine-point center. 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. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular: 
Title: Equilateral triangle
Content:  incircle). The integer-sided equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees. 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). Equilateral triangles are found in many other geometric constructs. 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. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles. In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three-dimensional analogue of the shape. The plane can be tiled using equilateral triangles giving the triangular tiling.
Title: Isosceles triangle
Content:  any triangle goes through the triangle's orthocenter (the intersection of its three altitudes), its centroid (the intersection of its three medians), and its circumcenter (the intersection of the perpendicular bisectors of its three sides, which is also the center of the circumcircle that passes through the three vertices). In an isosceles triangle with exactly two equal sides, these three points are distinct, and (by symmetry) all lie on the symmetry axis of the triangle, from which it follows that the Euler line coincides with the axis of symmetry. The incenter of the triangle also lies on the Euler line, something that is not true for other triangles. If any two of an angle bisector, median, or altitude coincide in a given triangle, that triangle must be isosceles.

Question: how many lines of symmetry are there in a equilateral triangle?
Answer: 3.