Elementary Geometry

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Equilateral Triangle

Side of an equilateral triangle: \(a\)
Angle of an equilateral triangle: \(\alpha = 60^\circ\)
Perimeter: \(P\)
Altitude: \(h\)
Radius of the circumscribed circle: \(R\)
Radius of the inscribed circle: \(r\)
Area: \(S\)
  1. An equilateral triangle is a triangle in which all three sides are equal. All angles in an equilateral triangle are equal to \(60^\circ\).
  2. Equilateral triangle
  3. In an equilateral triangle, the altitude, angle bisector, median and perpendicular bisector drawn from any vertex coincide.
  4. Relationship between the altitude (median, angle bisector or perpendiculr bisector) and the side
    \(h = {\large\frac{{a\sqrt 3 }}{2}\normalsize}\)
  5. Radius of the circumscribed circle (circumradius) of an equilateral triangle
    \(R = {\large\frac{{2h}}{3}\normalsize} = {\large\frac{{a\sqrt 3 }}{3}\normalsize}\)
  6. Radius of the inscribed circle (inradius) of an equilateral triangle
    \(r = {\large\frac{{h}}{3}\normalsize} = {\large\frac{{a\sqrt 3 }}{6}\normalsize}\)
  7. Relation between the circumradius and inradius in an equilateral triangle
    \(R = 2r\)
  8. Perimeter of an equilateral triangle
    \(P = 3a = 6\sqrt 3 r =\) \(3\sqrt 3 R\)
  9. Area of an equilateral triangle
    \(S = {\large\frac{{ah}}{2}\normalsize} = {\large\frac{{{a^2}\sqrt 3 }}{4}\normalsize} =\) \({\large\frac{{3{R^2}\sqrt 3 }}{4}\normalsize} =\) \(3\sqrt 3 {r^2}\)