Formulas

Elementary Geometry

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Quadrilateral

    • Sides of a quadrilateral: \(a,\) \(b,\) \(c,\) \(d\)
    Internal angles: \(\alpha\), \(\beta,\) \(\gamma,\) \(\delta\)
    Diagonals of a quadrilateral: \({d_1},\) \({d_2}\)
    Angle between the diagonals: \(\varphi\)
    Area: \(S\)
    Radius of the circumscribed circle: \(R\)
    Radius of the inscribed circle: \(r\)
    Perimeter: \(P\)
    Semiperimeter: \(p\)
    1. A quadrilateral is a polygon with four sides and four vertices (angles).
      2. Quadrilateral
    2. Types of quadrilaterals
      A quadrilateral with two pairs of parallel sides is called a parallelogram.
      A quadrilateral in which at least one pair of opposite sides is parallel is called a trapezoid.
      A quadrilateral in which all four angles are right angles is called a rectangle.
      A quadrilateral in which all four sides are equal is called a rhombus.
      A quadrilateral in which all four sides are equal and all four angles are right angles is called a square.
    3. Sum of the angles in a quadrilateral is \(360^\circ:\)
      \(\alpha + \beta + \gamma + \delta \) \(= 360^\circ\)
    4. Perimeter of a quadrilateral
      \(P = a + b \) \(+\; c + d\)
    5. Area of a quadrilateral
      \(S = {\large\frac{1}{2}\normalsize}{d_1}{d_1}\sin \varphi ,\)
      where \({d_1}\) and \({d_2}\) are the diagonals of the quadrilateral and \(\varphi\) is the angle between them.
    6. If the sums of opposite angles in a convex quadrilateral are equal to \(180^\circ,\) then the quadrilateral is called a cyclic quadrilateral. All four vertices of a cyclic quadrilateral lie on a circumscribed circle.
      \(\alpha + \beta = \gamma + \delta \) \(= 180^\circ\)
      7. Cyclic quadrilateral
    7. Ptolemy’s theorem
      In a cyclic quadrilateral, the sum of the products of opposite sides is equal to the product of the diagonals:
      \(ac + bd = {d_1}{d_2}\)
    8. Radius of the circumscribed circle
      \(R =\) \({\large\frac{1}{4}\normalsize}\sqrt {\large\frac{{\left( {ac + bd} \right)\left( {ad + bc} \right)\left( {ab + cd} \right)}}{{\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)\left( {p – d} \right)}}\normalsize} ,\)
      where \(p = {\large\frac{P}{2}\normalsize}\) is the semiperimeter of the quadrilateral.
    9. If the sums of opposite sides in a convex quadrilateral are equal, then the quadrilateral is called a tangential quadrilateral. All four sides of a tangential quadrilateral are tangents to an inscribed circle.
      \(a + d = b + c\)
      10. Tangential quadrilateral
    10. Radius of the inscribed circle
      \(r =\) \({\large\frac{{\sqrt {d_1^2d_2^2 – {{\left( {a – b} \right)}^2}{{\left( {a + b – p} \right)}^2}} }}{{2p}}\normalsize},\)
      where \(p = {\large\frac{P}{2}\normalsize}\) is the semiperimeter of the quadrilateral.