Formulas and Tables

Elementary Geometry

Scalene Triangle

Sides of a triangle: \(a\), \(b\), \(c\)
Angles of a triangle: \(A = \alpha,\) \(B = \beta,\) \(C = \gamma\)
Altitudes to the sides \(a\), \(b\), \(c\): \({h_a},\) \({h_b},\) \({h_c}\)
Medians to the sides \(a\), \(b\), \(c\): \({m_a},\) \({m_b},\) \({m_c}\)
Bisectors of the angles \(\alpha\), \(\beta\), \(\gamma\): \({t_a},\) \({t_b},\) \({t_c}\)

Midline of a triangle: \(q\)
Radius of circumscribed circle: \(R\)
Radius of inscribed circle: \(r\)
Semiperimeter of a triangle: \(p\)
Area: \(S\)

  1. A triangle is a geometric shape formed by three line segments that join three points not lying on a single line.
A scalene triangle
  1. Sum of the angles in a triangle is \(180^\circ\):
    \(\alpha + \beta + \gamma = 180^\circ\)
  2. Triangle inequality
    \(a + b \gt c\)
    \(b + c \gt a\)
    \(a + c \gt b\)
  3. \(\left| {a – b} \right| \lt c\)
    \(\left| {b – c} \right| \lt a\)
    \(\left| {a – c} \right| \lt b\)
  4. Midline of a triangle
    \(q = a/2,\;\;q\parallel a\)
Midline of a triangle
  1. Law of Cosines
    \({a^2} = {b^2} + {c^2} \) \(-\; 2bc\cos \alpha\)
    \({b^2} = {a^2} + {c^2} \) \(-\; 2ac\cos \beta \)
    \({c^2} = {a^2} + {b^2} \) \(-\; 2ab\cos \gamma \)
  2. Law of Sines
    \({\large\frac{a}{{\sin \alpha }}\normalsize} = {\large\frac{b}{{\sin \beta }}\normalsize} =\) \({\large\frac{c}{{\sin \gamma }}\normalsize} \) \(= 2R,\)
    where \(R\) is the radius of the circumscribed circle.
  3. Radius of the circumscribed circle
    \(R = {\large\frac{a}{{\sin \alpha }}\normalsize} = {\large\frac{b}{{\sin \beta }}\normalsize} =\) \({\large\frac{c}{{\sin \gamma }}\normalsize} =\) \({\large\frac{{bc}}{{2{h_a}}}\normalsize} =\) \({\large\frac{{ac}}{{2{h_b}}}\normalsize} =\) \({\large\frac{{ab}}{{2{h_c}}}\normalsize} =\) \({\large\frac{{abc}}{{4S}}\normalsize}\)
  4. Radius of the inscribed circle
    \({r^2} = {\large\frac{{\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)}}{p}\normalsize},\) \({\large\frac{1}{r}\normalsize} = {\large\frac{1}{{{h_a}}}\normalsize} + {\large\frac{1}{{{h_b}}}\normalsize} + {\large\frac{1}{{{h_c}}}\normalsize}\)
  5. Finding the angles of a triangle given \(3\) sides
    \(\sin {\large\frac{\alpha }{2}\normalsize} = \sqrt {\large\frac{{\left( {p – a} \right)\left( {p – c} \right)}}{{bc}}\normalsize} \)
    \(\cos{\large\frac{\alpha }{2}\normalsize} = \sqrt {\large\frac{{p\left( {p – a} \right)}}{{bc}}\normalsize} \)
    \(\tan{\large\frac{\alpha }{2}\normalsize} = \sqrt {\large\frac{{\left( {p – b} \right)\left( {p – c} \right)}}{{p\left( {p – a} \right)}}\normalsize} \)
  6. Finding the altitudes of a triangle given \(3\) sides
    \({h_a} =\) \({\large\frac{2}{a}\normalsize}\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)} \)
    \({h_b} =\) \({\large\frac{2}{b}\normalsize}\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)} \)
    \({h_c} =\) \({\large\frac{2}{c}\normalsize}\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)} \)
  7. Finding the altitudes of a triangle given a side and an angle
    \({h_a} = b\sin \gamma = c\sin \beta \)
    \({h_b} = a\sin \gamma = c\sin \alpha \)
    \({h_c} = a\sin \beta = b\sin \alpha \)
  8. Finding the medians of a triangle given \(3\) sides
    \(m_a^2 = {\large\frac{{{b^2} + {c^2}}}{2}\normalsize} – {\large\frac{{{a^2}}}{4}\normalsize}\)
    \(m_b^2 = {\large\frac{{{a^2} + {c^2}}}{2}\normalsize} – {\large\frac{{{b^2}}}{4}\normalsize}\)
    \(m_c^2 = {\large\frac{{{a^2} + {b^2}}}{2}\normalsize} – {\large\frac{{{c^2}}}{4}\normalsize}\)
Finding the medians of a triangle given 3 sides
  1. Distance from a vertex to the point of intersection of the medians (centroid)
    \(AM = {\large\frac{2}{3}\normalsize}{m_a},\;\) \(BM = {\large\frac{2}{3}\normalsize}{m_b},\;\) \(CM = {\large\frac{2}{3}\normalsize}{m_c}\)
  2. Finding the bisectors given 3 sides
    \(t_a^2 = {\large\frac{{4bcp\left( {p – a} \right)}}{{{{\left( {b + c} \right)}^2}}}\normalsize}\)
    \(t_b^2 = {\large\frac{{4acp\left( {p – b} \right)}}{{{{\left( {a + c} \right)}^2}}}\normalsize}\)
    \(t_c^2 = {\large\frac{{4abp\left( {p – c} \right)}}{{{{\left( {a + b} \right)}^2}}}\normalsize}\)
  3. Area of a triangle
    \(S = {\large\frac{{a{h_a}}}{2}\normalsize} = {\large\frac{{b{h_b}}}{2}\normalsize} = {\large\frac{{c{h_c}}}{2}\normalsize}\)
    \(S = {\large\frac{{ab\sin \gamma }}{2}\normalsize} = {\large\frac{{ac\sin \beta }}{2}\normalsize} =\) \({\large\frac{{bc\sin \alpha }}{2}\normalsize}\)
    \(S = pr\)
    \(S = {\large\frac{{abc}}{{4R}}\normalsize}\)
    \(S = 2{R^2}\sin \alpha \sin \beta \sin \gamma \)
    \(S = {p^2}\tan{\large\frac{\alpha }{2}\normalsize} \tan{\large\frac{\beta }{2}\normalsize} \tan{\large\frac{\gamma }{2}\normalsize}\)
  4. Heron’s formula
    \(S =\) \(\sqrt {p\left( {p – a} \right)\left( {p – b} \right)\left( {p – c} \right)} \)
    where \(p\) is the semiperimeter of the triangle.