Mathematics — Euclid-Heron area of a triangle

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After introducing the height of a triangle in the spirit of Pythagoras theorem, we compute the height and we obtain the Heron formula for the area of a triangle.

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Quadratic excess  &  3 kinds of triangles

We define the quadratic excess of a real ordered triple (a b c) as the excess of the squared triple (a2 b2 c2):

qxc(a b c)  :=  exc(a2 b2 c2)  =  (b2 + c2 – a2) / 2

Now let T  :=  (a b c) ε T is a triangle. Then we also define:

  • qa  :=  qxc(T)  =  (b2 + c2 – a2) / 2
  • qb  :=  qxc(lft(T))  =  (a2 + c2 – b2) / 2
  • qc  :=  qxc(rgt(T))  =  (a2 + b2 – c2) / 2

When qxt(T) > 0 then we say that the front angle of T is acute, when qxt(T) = 0 then the front angle of T is called right, and when qxt(T) < 0 then obtuse. The front angle of T is right if and only if the Pythagoras equality holds:

a2  =  b2 + c2

The front angles (whatever it means) of triangles lft(T) and rgt(T) are considered to be the other two angles of T.

THEOREM 0: Let T := (a b c) ε T be an arbitrary triangle. Then at least two of the angles of T are acute, i.e. at most one of the coefficients qa qb qc is not positive.

PROOF: The sum of any two of the quadratic excess coefficients is positive:

  • qa + qb = c2 > 0
  • qa + qc = b2 > 0
  • qb + qc = a2 > 0

End of proof.

If a triangle has an obtuse angle then it is called an obtuse triangle; if it has a right angle then it is called a right triangle; otherwise, in the remaining case, it is called an acute triangle. Let Act be the set (space) of all acute triangles. Then mappings art : T –> Act (acute root) and asq : Act –> T (acute squared), given by the formulas:

  • art (a b c)  :=  (√a √b √c)     for every (a b c) ε T;
  • asq (a b c)  :=  (a2 b2 c2)     for every (a b c) ε Art;

are bijections, each inverse to the other.

Similar triangles

Triangles T := (a b c) and T’ := (a’ b’ c’) are called similar if there is a real number t such that

(a’ b’ c’)  =  t ⋅ (a b c)

i.e.

a’  =  t⋅a   and   b’  =  t⋅b   and   c’  =  t⋅c

Such t must be positive It is called the coefficient of the similarity. Observe that similar triangles are of the same type (acute, right or obtuse), and their respective angles are respectively of the same type. Furthermore, mappings art and asq preserve similarity–similar triangles T and T’ are mapped into a pair of similar triangles (but the similarity coefficient t is replaced by √t or by t2 respectively).

Definition and computation of the height

DEFINITION 0: The front height h := hgt(T) of triangle T := (a b c) is defined as a non-negative real number which, together with two other real numbers ab ac, satisfy the following three equalities:

  • ab + ac  =  a
  • h2 + ab2  =  b2
  • h2 + ac2  =  c2

Subtract the last equality from the second one:

(ab – ac)⋅a  =  b2 – c2

hence

ab – ac  =  (b2 – c2) / a;
2⋅ab – a  =  (b2 – c2) / a

and

ab  =  (a2 + b2 – c2) / (2⋅a)  =  qc/a;

also (along a similar line)

ac  =  (a2 + c2 – b2) / (2⋅a)  =  qb/a.

Clearly, the first equality of Definition 0, ab + ac  =  a, holds. Next:

h2  =  b2 – ab2  =  (b + ab)⋅(b – ab)
=  ((a+b)2 – c2)⋅(c2 – (a-b)2) / (4⋅a2)
=  (a+b+c)⋅(a+b-c)⋅(a+c-a)⋅(b+c-a) / (4⋅a2)
=  4 ⋅ p(T) ⋅ ea ⋅ eb ⋅ ec / a2

hence

hgt(T)  =  (2/a)⋅√ (p(T)⋅ea⋅eb⋅ec)

Thus the height of a triangle is always positive,

hgt(T)  >  0

We see that if  h ab ac exist then they are unique and given by the formulas above. We also know that the first equality of Definition 0 is satisfied. But we saw above that  h2  =  b2 – ab2,  hence the second equality of Definition 0 holds as well.  The third equality holds too due to the symmetry of the formula for hgt(T):

hgt(tra(T))  =  hgt(T)

Thus the triple  h ab ac  exists, is unique, and given by the above formulas.

Heron formula for area of a triangle

The euclidean area of a triangle T := (a b c) is defined as

Heron(T)  :=  a⋅hgt(T) / 2.

To avoid being dependent on the abc-notation, let’s introduce:

edge(T)  :=  a

to be the front edge of a triangle. Now we can say:

DEFINITION 1:

Heron(T)  :=  edge(T)⋅hgt(T) / 2

The formula for hgt(T) from the previous section gives us instantly the Heron formula:

THEOREM 1:

Heron(T)  =  √ (p(T)⋅ea⋅eb⋅ec)
 
The first impression from Definition 1 (above) is that the area depends on the ordering of the edges. However, the Heron formula shows that area does not depend on the order of edges a b c, i.e.

THEOREM 2:

Heron(T)  =  Heron(lft(T))  =  Heron(rgt(T))  =
Heron(tra(T))  =  Heron(trb(T))  =  Heron(trc(T))

Let us recall that triangles  T := (a b c)  and  T’ := (a’ b’ c’)  are called similar  ⇐:⇒  there exists a real  t > 0  such that  T’ = t⋅T,  i.e.  a’ = t⋅a  and  b’ = t⋅b  and  c’ = t⋅c.  Obviously:

Heron(t⋅T)  =  t2⋅Heron(T)
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