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Angle Measures (Posted on 2006-07-05) Difficulty: 3 of 5
What are the measures of the apex angles of all isosceles triangles with circumradius 9 and inradius 4?

(As a reminder, the circumradius is the radius of the circumscribed circle, and the inradius is the radius of the inscribed circle.)

  Submitted by Bractals    
Rating: 2.0000 (2 votes)
Solution: (Hide)


There are only two. The smaller apex angle (saa) occurs when the circumcenter lies on the line segment joining the apex angle with the incenter. The larger apex angle (laa) occurs when the incenter lies on the line segment joining the apex angle with the circumcenter. The distance between the circumcenter and incenter is
         d = sqrt(R(R-2r)) = 3
Therefore,
       saa = 2 arcsin(IS'/(SO + OI))

           = 2 arcsin(r/(R + d))

           = 2 arcsin(1/3)

           ~= 38.94244 degrees

       laa = 2 arcsin(IL'/(LO - OI))

           = 2 arcsin(r/(R - d))

           = 2 arcsin(2/3)

           ~= 83.62063 degrees
where S and L are the vertices at the smaller and larger apex angles, S' and L' are the feet of the corresponding altitudes, and I and O are the incenter and circumcenter respectively.

See Richard's "Using a Formula" for an alternate method.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: An almost complete estimated solution (spoiler)Richard2006-07-06 17:12:09
SolutionAn almost complete estimated solution (spoiler)Grate Guy2006-07-06 16:16:48
Using a FormulaRichard2006-07-05 22:00:40
SolutionA jump to the peak (spoiler)Dej Mar2006-07-05 19:45:07
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