what are the properties of a equilateral triangle
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all the Side are equal
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Here are some properties of equilateral triangles.
In an equilateral triangle, all three sides are equal, by definition.
In Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.
An equilateral triangle is a regular polygon, so it has all the properties of regular polygons. For example, since a regular -gon has lines of symmetry, an equilateral triangle has lines of symmetry.
The Altitude, median, angle bisector, and the perpendicular bisector of a given side are all the same line, and is one of the three lines of symmetry of the triangle.
The orthocenter, circumcenter, incenter,[ centroid, and nine-point center are all the same point, so the Euler line is not defined for an equilateral triangle.
The altitude of an equilateral triangle is where is the side length, so the area of an equilateral triangle is
The circumradius, (radius of the circumcircle) of an equilateral triangle is which is of the altitude.
The inradius, (radius of the incircle) of an equilateral triangle is which is of the altitude. This is also the apothem of the regular -gon, or perpendicular distance from the center to a side.
By Euler's inequality, the equilateral triangle has the smallest ratio of the circumradius to the inradius of any triangle: specifically,
The exradius (radius of an excircle) of an equilateral triangle is which is equal to the altitude because the excircle can be circumscribed by a hexagon made from six equilateral triangles with side which you can see using the GeoGebra applet, Equilateral Triangles Incircle Circumcircle Excircle.
Viviani’s theorem states: The sum of the perpendicular distances from an interior point, P, to each of the sides is equal to the altitude of the equilateral triangle.
Napoleon's theorem states: if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle, which is called a Napoleon triangle. The centers of both the inner and outer Napoleon triangles coincide with the centroid of the original triangle.
If a triangle is placed in the complex plane with complex vertices and then for either non-real cube root of the triangle is equilateral if and only if I made a GeoGebra applet called the Equilateral Triangle Polynomial Test, which illustrates geometrically how the vertices of equilateral triangle ABC are rotated through angles of 120°, 240°, and 0°, respectively, to form triangle A’B’C whose centroid is at the origin.
For any point on the minor arc of the circumcircle, with distances and from and respectively, This is a fascinating result, which I proved here.
Equilateral triangles are the only triangles whose Steiner inellipse and Steiner circumellipse are circles, which are the incircle and circumcircle, respectively. See the GeoGebra applet called Steiner Ellipses.
In an equilateral triangle, all three sides are equal, by definition.
In Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.
An equilateral triangle is a regular polygon, so it has all the properties of regular polygons. For example, since a regular -gon has lines of symmetry, an equilateral triangle has lines of symmetry.
The Altitude, median, angle bisector, and the perpendicular bisector of a given side are all the same line, and is one of the three lines of symmetry of the triangle.
The orthocenter, circumcenter, incenter,[ centroid, and nine-point center are all the same point, so the Euler line is not defined for an equilateral triangle.
The altitude of an equilateral triangle is where is the side length, so the area of an equilateral triangle is
The circumradius, (radius of the circumcircle) of an equilateral triangle is which is of the altitude.
The inradius, (radius of the incircle) of an equilateral triangle is which is of the altitude. This is also the apothem of the regular -gon, or perpendicular distance from the center to a side.
By Euler's inequality, the equilateral triangle has the smallest ratio of the circumradius to the inradius of any triangle: specifically,
The exradius (radius of an excircle) of an equilateral triangle is which is equal to the altitude because the excircle can be circumscribed by a hexagon made from six equilateral triangles with side which you can see using the GeoGebra applet, Equilateral Triangles Incircle Circumcircle Excircle.
Viviani’s theorem states: The sum of the perpendicular distances from an interior point, P, to each of the sides is equal to the altitude of the equilateral triangle.
Napoleon's theorem states: if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle, which is called a Napoleon triangle. The centers of both the inner and outer Napoleon triangles coincide with the centroid of the original triangle.
If a triangle is placed in the complex plane with complex vertices and then for either non-real cube root of the triangle is equilateral if and only if I made a GeoGebra applet called the Equilateral Triangle Polynomial Test, which illustrates geometrically how the vertices of equilateral triangle ABC are rotated through angles of 120°, 240°, and 0°, respectively, to form triangle A’B’C whose centroid is at the origin.
For any point on the minor arc of the circumcircle, with distances and from and respectively, This is a fascinating result, which I proved here.
Equilateral triangles are the only triangles whose Steiner inellipse and Steiner circumellipse are circles, which are the incircle and circumcircle, respectively. See the GeoGebra applet called Steiner Ellipses.
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