Geometry, Online Education

Heron's Formula, Triangle Area. Step by Step Proof

If triangle ABC has sides a, b, c and semi perimeter \(s=\dfrac{a+b+c}{2}\) then area of triangle ABC is \(K=\sqrt {s(s-a)(s-b)(s-c)}\)

Proof

  1. By propositions #5, and #6: D Incenter and E excenter

  2. By proposition #8: \(AF= s-a\) and \(CF= s-c\)

  3. By proposition #9: \(CG= s-b\)

  4. By proposition #11 Area ABC: \(K= r\cdot s\)

  5. By proposition #12 Area ABC: \(K= r_a(s-a)\)

  6. By multiplying (4) and (5):  \(K^2= r\cdot r_a\cdot s(s-a)\) 

  7. By propositions #13, #14, #15 triangles DFC and CGE are similar (Similarity AA)
    Therefore \(\dfrac{r}{s-b}=\dfrac{s-c}{r_a}\implies r\cdot r_a=(s-b)(s-c)\)

  8. Substituting in (6): \(K^2= s(s-a)(s-b)(s-c)\).
    Therefore \(K=\sqrt {s(s-a)(s-b)(s-c)} \text{    Q.E.D.}\)

 

See also:

Heron's Formula Proof Mind Map
Heron's Formula: Key facts, preliminary propositions.  


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Last updated: November 18, 2007