Call the center of the inscribed circle O and the sides a,b,c. Then r is the altitude for triangle OAB with base c which has area = r*c/2 and the area of ABC = r*(a+b+c)/2.
But the area of ABC can also be expressed as h(C)*c/2.
Equating and simplifying gives 1/h(C) = c*(1/r)*(1/(a+b+c).
Doing the same for the other two triangles and adding gives
1/h(A) + 1/h(B) + 1/h(C) = (a+b+c)*(1/r)*(1/(a+b+c) = 1/r.
That matches the formula following 'i.e.' in the problem but not the preceding text which I believe should read 'the inradius equals the reciprocal of 1/3 the harmonic average of the 3 altitudes'.
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Posted by xdog
on 2020-05-08 08:52:40 |
solution and correction
