[alogo] 1. Fundamental invariants of triangle ABC

These are the three quantities associated with a triangle ABC:
s = the half-perimeter  s = (a+b+c)/2,
r = the inradius i.e. the radius of inscribed circle
R = the circumradius i.e. the radius of the circumcircle of the triangle.

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[alogo] 2. Some remarkable identities

Denoting by {ra, rb, rc} the radii of the excircles of the triangle the following identity is valid [Johnson, p.189]:

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The proof relies on some other identities involving the area Delta of the triangle and the quantities s, s-a,
s-b, s-c (see Bisector1.html ):

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The first expresses the radii in terms of the area and the perimeter. The second sums over the cyclic permutations
of the letters {a,b,c,}

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Last equation results by carrying out the operations (e.g. with Maxima). Then back substitution yields

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This, taking into account Heron's formula for the area, and last expressing a, b, c in terms of sines by the sine
formula giving:

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By the occasion of this calculation I include another couple of formulas involving the two symmetric
quadratic expressions of the sides of the triangle.

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Denote the first sum by X and the second  by Y. Obviously
                                                 2X + Y  =  (a+b+c)2  =  4s2.
On the other side the expression Y-2X can be written:

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Replacing there s-a, s-b, s-c with the expressions resulting from 1. we obtain:

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Solving these formulas for {X,Y} we find the expressions in 7. and 8.

[alogo] 3. Generalizing

The previous method can be generalized to compute every symmetric function of {a,b,c} in terms of the
distinguished quantities {s, R, r} (perimeter, circumradius, inradius), called fundamental invariants of the triangle
ABC (see [Andreescu, p. 110]) . As an example I examine the two basic
cubic symmetric functions:  X = (a3+b3+c3) and  Y = (bc(b+c)+ca(c+a)+ab(a+b). They satisfy:

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Solving the two equations for {X,Y} we find the expressions for these two symmetric cubic functions.

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The calculation of the higher symmetric functions has to be done gradually, since in each step the results of
the previous steps are needed.
A use of these formulas is made in the GIO construction problem i.e. the problem of constructing a triangle
by giving the location of its three remarkable points: G(centroid), I(incenter) and O(circumcenter). This is
discussed in GIO_Construction.html .

As a last example I calculate the symmetric functions of fourth order:

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[1_0] [1_1] [1_2] [1_3]
[2_0] [2_1] [2_2] [2_3]

See Also

Bisector1.html
GIO_Construction.html

Bibliography

[Andreescu] Titu Andreescu and Dorin Andrica Complex Numbers from A to Z Boston, Birkhaeuser 2005
[Johnson] Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929

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