Incircles and Excircles in a Triangle
The points of tangency of the incircle of triangle ABC with sides a, b, c, and semiperimeter
The length of those tangents from the vertices of the triangle to its incircle can be easily determined. Denote them x, y, z, as in the diagram. We have three equations:
(1) |
x + y = c y + z = a z + x = b, |
from where
(2) | x + y + z = p. |
Subtracting from (2) equations (1) one at a time, we get
(3) |
x = p - a y = p - b z = p - c. |
These are the lengths that appear in Heron's formula, e.g.,
Similarly we can find the lengths of the tangents to the excircles.
Obviously, 2p = (b + u) + (c + v). But, since
(4) |
u = p - b v = p - c. |
From the just derived formulas it follows that the points of tangency of the incircle and an excircle with a side of a triangle are symmetric with respect to the midpoint of the side. Such points are called isotomic. The cevians joinging the two points to the opposite vertex are also said to be isotomic. Both triples of cevians meet in a point. For the incircle, the point is Gergonne'; for the points of excircle tangency, the point is Nagel's. We have just proved that, in any triangle, the Gergonne and Nagel points are isotomic conjugate of each other. (This fact has an interesting geometric illustration.)
In general, two points in a triangle are isotomic conjugate if the cevians through them are pairwise isotomic. The centroid is one point that is its own isotomic conjugate.
S = ra(p - a)
Indeed,
2S | = (b + u)ra + (c + v)ra - ara - (u + v)ra |
= (b + c - a)ra | |
= (2p - 2a)ra | |
= 2(p - a)ra. |
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