Incircle and Excircles of A Triangle - Relation To Area of The Triangle

Relation To Area of The Triangle

The radii of the in- and excircles are closely related to the area of the triangle. Let K be the triangle's area and let a, b and c, be the lengths of its sides. By Heron's formula, the area of the triangle is


\begin{align}
K & {} = \frac{1}{4}\sqrt{(P)(a-b+c)(b-c+a)(c-a+b)} \\
& {} = \sqrt{s(s-a)(s-b)(s-c)}
\end{align}

where is the semiperimeter and P = 2s is the perimeter.

The radius of the incircle (also known as the inradius, r ) is

Thus, the area K of a triangle may be found by multiplying the inradius by the semiperimeter:

The radii in the excircles are called the exradii. The excircle at side a has radius

Similarly the radii of the excircles at sides b and c are respectively

and

From these formulas one can see that the excircles are always larger than the incircle and that the largest excircle is the one tangent to the longest side and the smallest excircle is tangent to the shortest side. Further, combining these formulas with Heron's area formula yields the result that

The ratio of the area of the incircle to the area of the triangle is less than or equal to, with equality holding only for equilateral triangles.

Read more about this topic:  Incircle And Excircles Of A Triangle

Famous quotes containing the words relation to, relation and/or area:

    Concord is just as idiotic as ever in relation to the spirits and their knockings. Most people here believe in a spiritual world ... in spirits which the very bullfrogs in our meadows would blackball. Their evil genius is seeing how low it can degrade them. The hooting of owls, the croaking of frogs, is celestial wisdom in comparison.
    Henry David Thoreau (1817–1862)

    To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.
    Henry James (1843–1916)

    Many women are reluctant to allow men to enter their domain. They don’t want men to acquire skills in what has traditionally been their area of competence and one of their main sources of self-esteem. So while they complain about the male’s unwillingness to share in domestic duties, they continually push the male out when he moves too confidently into what has previously been their exclusive world.
    Bettina Arndt (20th century)