Contact triangles appear in triangle and circle geometry.

An inscribed circle has a contact triangle associated with it. The inscribed circle touches the interior of the triangle at three points, which forms the contact triangle's vertices. The three contact points, TA, TB, and TC are the points at which the incircle touches the reference triangle ΔABC. The contact triangle ΔTATBTC is sometimes also referred to as the Gergonne triangle.

The contact triangle's side lengths are:

     a' = (-a+b+c)⋅cos(α/2)                                        (1a)
     b' = ( a-b+c)⋅cos(β/2)                                        (1b)
     c' = ( a+b-c)⋅cos(γ/2)                                        (1c)

where a, b, and c are the sidelengths of the reference triangle and α, β, and γ are the interior angles.

The area of the contact triangle is related to the area of the reference triangle and, it turns out, to the radius of the inscribed or circumscribed circle - whichever is most convenient. The area can be calculated using any of the following expressions:


 K'    r
 -  = ---                                                         (2a)
 K    2⋅R

       K   (-a+b+c)⋅(a-b+c)⋅(a+b-c)
 K' =  - * ----------------------                                 (2b)
       4        (a⋅b⋅c)

        2⋅K
    = ------ * s⋅r2                                                (2c)
      (a⋅b⋅c)

       K2
    = ----                                                        (2d)
      2⋅s⋅R

where r is the radius of the incircle, R is the radius of the circumcircle, and s is the semiperimeter of the reference triangle.

There are three more contact triangles, called excontact triangles - one for every excircle. Expressions for their sidelengths and areas are also available, but uninteresting.

A relationship between areas is interesting.

The ratio of the sums of the areas of the three excontact triangles minus the reference triangle's contact triangle to the area of the reference triangle is always equal to two. In other words, the sum of the areas of the excontact triangles KA+KB+KC minus the area of the contact triangle K' (as found above) is equal to twice the area of the reference triangle, K.

  2K = KA+KB+KC - K'                                                (3)

The ratio of areas of an excontact triangle to the contact triangle is equal to the ratio of the radius of the incircle to the radius of the excircle. (It's interesting that it's the ratio of the radii, and not the ratio of the radii squared!)

  
   K'   r
  --- = --                                                         (4)
   KA    rA

Another relationship between areas is that the reciprocal of the area of the contact triangle is equal to the sum of the reciprocals of the areas of the three excontact triangles, well illustrated here.

    1     1     1     1
   --- = --- + --- + ---                                           (5)
    K'    KA    KB    KC

The reference triangle ΔABC actually has six contact points touching it. The three contact points, TA, TB, and TC are the points at which the incircle touches ΔABC. The contact triangle described so far is the triangle ΔTATBTC. There are three other contact points on the reference triangle, TA', TC', and TC' 1. This triple of contact points also has a contact associated with it, ΔTA'TB'TC' (illustration here), and it has the same area as the other contact triangle.

A drawing of the reference triangle's contact triangle and the three excontact triangles appears on my homenode, and will remain there for a few days.



NOTES

  1. You may be interested to know that the contact points TA and TA', both located on side a, are symmetric reflections about the midpoint of the side, mA, that is, the distance between mA and TA is the same as between mA and TA'. The same is true for all three sides.

Internet References

  1. Weisstein, Eric W. "Contact Triangle" From MathWorld--A Wolfram Web Resource.
  2. Wikipedia, "Incircle and Excircles of a Triangle"
  3. Antonio Guiterrez, Problem 82, "Area of a Contact Triangle, Inradius, circumradius," GoGeometry. The ratio of the area of the contact triangle to the reference triangle is equal to the ratio of the incircle's radius to twice the circumcenter's radius. (Equation 2a)
  4. Antonio Guiterez, Problem 84, "Contact Triangles Areas, Incircle, Excircle," GoGeometry. The ratio of the area of the contact triangle to excontact triangle A is equal to the ratio of the radius of the incircle to the radius of the excircle A. (Equation 4)
  5. Antonio Guiterez, Problem 85, "Contact Triangles, Incircle, Excircle," GoGeometry. The reciprocal of the area of the contact triangle is equal to the sum of the reciprocals of the areas of the three excontact triangles. (Equation 5)
  6. Antonio Guiterez, Problem 86, "Intouch and Extouch Triangles, Areas," GoGeometry. The area of contact triangle ΔTATBTC is the same as the area of its dual, the excontact triangle ΔTA'TB'TC'.

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