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 Heron's Formula. Step by Step Proof.

If triangle ABC has sides a, b, c and semiperimeter then area of triangle ABC
 

 

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:

  5. By proposition #12 Area ABC:

  6. By multiplying (4) and (5): 

  7. By propositions #13, #14, #15 (Similarity AA):
    Therefore

  8. Substituting in (6):
    Q.E.D.

 

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Last updated: February 14, 2007