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Little explorations with HP calculators (no Prime)
04-09-2017, 03:40 PM (This post was last modified: 04-09-2017 03:43 PM by Gerson W. Barbosa.)
Post: #166
RE: Little explorations with the HP calculators
(04-08-2017 09:17 PM)Gerson W. Barbosa Wrote:    Sabc                   a b c
 ------ = -----------------------------------
  Sdef    a(b - r)(c - s) + t(c r + b(s - c))


Making r = b/2, s = c/3 and t = a/4 per the original problem, the expression easily evaluates to 24/7. Perhaps r should be associated with side a, s to side b, t to side c and the expression rewritten accordingly for mnemonic purposes.

A more compact formula in terms of the relationships between the segments and the sides of the outer triangle is better:

Given the triangles ABC and DEF, where

BC = a
BF = r = a/x
CF = a - r

AC = b
CE = s = b/y
AE = b - s

AB = c
AD = t = c/z
BD = c - t


Then
 
  SΔABC              x y z
 ------ = --------------------------
  SΔDEF     z + (x y - x - y)(z - 1)



Examples:

1) x = 2, y = 3, z = 4

 SΔABC/SΔDEF = (2*3*4)/(4 + (2*3 - 2 - 3)*(4 - 1) = 24/7

2) x = 2, y = 3, z = 2

 SΔABC/SΔDEF = (2*3*2)/(2 + (2*3 - 2 - 3)*(2 - 1) = 4


[Image: S1overS2_formula_zpskhavaqhu.jpg]
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RE: Little explorations with the HP calculators - Gerson W. Barbosa - 04-09-2017 03:40 PM



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