Let ABC be an arbitrary triangle with side lengths a = |BC|, b = |CA|, and c = |AB|.
Let X, Y, and Z be the points of tangency of the incircle with the sides BC, CA, and AB respectively.
Let X', Y', and Z' be the points of tangency of the excircles with "sides" BC, CA, and AB respectively.
What is the value of |XX'| + |YY'| + |ZZ'| in terms of a, b, and c?
let t=|BX| by similar triangles we have |BZ|=|BX|=t we also have |CX|=|CY|=a-t |AZ|=|AY|=b-a+t thus |AZ|+|BZ|=|AB|
let Z,W be the centers of the excircle from side BC and incircle respectivly
let Rx and Ri be the radii of the excircle and incircle respectivly
ZBX' and BWX are right triangles and since angles BZX' and BWX are congruent and ZBX'=BXW=90 the we have that
now if A is the area of triangle ABC then
substituting in our know value for t we get
now using Herons formula for area we get
simplfying we get
now we can get |XX'| with |t-u| which is
now using similar methods we can get |YY'| and |ZZ'| to be |a-c| and |a-b| respectivly thus the desired sum is
Posted by Daniel
on 2008-01-18 02:26:11