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.
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):
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:
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.
Triangles T := (a b c) and T’ := (a’ b’ c’) are called similar if there is a real number t such that
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:
2⋅ab – a = (b2 – c2) / a
also (along a similar line)
Clearly, the first equality of Definition 0, ab + ac = a, holds. Next:
= ((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
Thus the height of a triangle is always positive,
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):
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
To avoid being dependent on the abc-notation, let’s introduce:
to be the front edge of a triangle. Now we can say:
The formula for hgt(T) from the previous section gives us instantly the Heron formula:
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.
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: