Scalene Triangle Formulas

Posted In: Mathematics

A scalene triangle is a triangle that has three unequal sides.

Scalene Triangle

Sides of triangle: a,b,c
Semiperimeter: $p={a+b+c}/2$
Angles of triangle: α,β,γ
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 α,β,γ: t_a,t_b,t_c
Radius of circumscribed circle: R
Radius of inscribed circle: r
Area: S

  1. $α+β+γ=180°$
  2. $a+b>c,$
    $b+c>a,$
    $c+a>b$
  3. $a-b<c,$
    $b-c<a,$
    $a-c<b$
  4. Midline
    $q=a/2, q‖a$
  5. Law of Cosines
    $a^2=b^2+c^2-2bc\ cos\ α$
    $b^2=a^2+c^2-2ac\ cos\ β$
    $c^2=a^2+b^2-2ab\ cos\ γ$
  6. Law of Sines
    $a/{sin\ α}=b/{sin\ β}=c/{sin\ γ}=2R$
    where R is the radius of the circumscribed circle.
  7. $R=a/{2sin\ α}=b/{2sin\ β}=c/{2sin\ γ}$
    $={bc}/{2h_a}={ac}/{2h_b}={ab}/{2h_c}={acb}/{4S}$
  8. $r^2={(p-a)(p-b)(p-c)}/p,$
    $1/r=1/h_a+1/h_b+1/h_c$
  9. $sin{α/2}=√{{(p-b)(p-c)}/{bc}}$
    $cos{α/2}=√{{p(p-a)}/{bc}}$
    $tan{α/2}=√{{(p-b)(p-c)}/{p(p-a)}}$
  10. $h_a=2/a√{p(p-a)(p-b)(p-c)}$
    $h_b=2/b√{p(p-a)(p-b)(p-c)}$
    $h_c=2/c√{p(p-a)(p-b)(p-c)}$
  11. $h_a=bsin\ γ=csin\ β$
    $h_b=asin\ γ=csin\ α$
    $h_c=asin\ β=bsin\ α$
  12. $m_a^2={b^2+c^2}/2-a^2/4$,
    $m_b^2={a^2+c^2}/2-b^2/4$,
    $m_c^2={a^2+b^2}/2-c^2/4$
  13. $AM=2/3m_a$
    $BM=2/3m_b$
    $CM=2/3m_c$
  14. $t_a^2={4bcp(p-a)}/(b+c)^2,$
    $t_b^2={4acp(p-b)}/(a+c)^2,$
    $t_c^2={4abp(p-c)}/(a+b)^2$
  15. $S={ah_a}/2={bh_b}/2={ch_c}/2,$
    $S={ab\ sin\ γ}/2={ac\ sin\ β}/2={bc\ sin\ α}/2,$
    $S=√{p(p-a)(p-b)(p-c)}/2,$ Heron’s Formula,
    $S=pr,$
    $S={abc}/{4R},$
    $S=2R^2sin\ α\ sin\ β\ sin\ γ,$
    $S=p^2tan{α/2}\ tan{β/2}\ tan{γ/2}$

Post Tags: Geometry


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