Angle Measures (Posted on 2006-07-05)

	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.

	Subject	Author	Date
	re: An almost complete estimated solution (spoiler)	Richard	2006-07-06 17:12:09
	An almost complete estimated solution (spoiler)	Grate Guy	2006-07-06 16:16:48
	Using a Formula	Richard	2006-07-05 22:00:40
	A jump to the peak (spoiler)	Dej Mar	2006-07-05 19:45:07