An Approach to the Triangle

In this Section, we shall develop a formula for expressing the { Area of the Triangle } in terms of the { lengths of its 3 sides } .

Let us first set-up , in general , a triangle :

• with { L1 } , { L2 } , & { L3 } being the lengths of its 3 sides , respectively , as per this diagram below :

Let us now set up the value { H } as the height of the triangle , as per this diagram below :

The area of the triangle , denoted by { A-O-T } , is then given by :

Let us also set up the lengths { X } & { Y } , as shown in the diagram above , so that :

And the { Pythagorus Equations } for the two (2) { Right-Angle Triangles } associated with { X } & { Y } , respectively , are then :

We shall now attempt to express the { Area of the Triangle } , in terms of { L1 } , { L2 } , & { L3 } .

Let us now take the 3 equations immediately above and re-express the last 2 equations as follows :

Substituting these back into the 1st equation then yields :

Squarimg both sides of the equation then yields :

Re-arranging terms then yields :

Squaring both sides then yields :

We then expand the { left-hand-side } of the equation to arrive at this :

Re-arranging , cancelling and collecting terms then yields :

Let us now bring back this equation for the { Area of the Triangle } from { Topic #2 } above :

Squaring both side and then multiplying both sides by { 16 } then yield the relation :

Substituting this back into the above equation then yields the formula :

And this is the formula we wanted .

An alternate format of this equation is then :

• 

  ( some readers may wish to check this at their own pace ) .

First of all , if { L1 = L2 = L3 = 1 } , we have an { equilateral triangle } whose sides are of { unit-lengths } :

• the value of ( 16 * { A-O-T }^2 ) is then equal to { 3 } .

And in general , whenever the values of { L1 } , { L2 } , & { L3 } are all { integers } :

• the value of ( 16 * { A-O-T }^2 ) is also an { integer } .

  But this { integer } cannot be a { prime number } , except in the special case of an { equilateral unit-length triangle } where the value is { 3 } , as noted immediately above .

• it also follows that :
  • { Sine-Square } & { Cosine-Square } for the 3 { angles } are { whole fractions } .

  And this will be investigated in Section IX - The { Sine-Square } & { Cosine-Square } Functions .

In the next Section , we shall attempt to attach a { geometric } significance to the value ( 4 * { A-O-T } ) , i.e. :

• the { square-root } of ( 16 * { A-O-T }^2 ) .