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Balf
Member since Jul-4-07
 Jul-04-07, 11:24 PM (EST)
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"Scalene triangle angle to side relationships"
 
   Hello. If a scalene triangle has a base with known length and one side with a length of one third of the other, is there any relationship between the angle between the base and the shorter of the other two sides?

Thanks,

Peter


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mpdlc
Member since Mar-12-07
 Jul-08-07, 04:00 PM (EST)
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1. "RE: Scalene triangle angle to side relationships"
In response to message #0
 
   With the scanty information given calling A and B the two endpoints of the base of our triangle and calling C the other vertex, it is obvious that C lays in a circumference (c) since the ratio of distances AC/BC = 3 .This circumference is symmetrical with respect to the line containing the base and we can obtain easily the two endpoints of its diameter M and N for the given ratio.

For example taking the origin at midpoint of the base If we allocate coordinates to A and B we will get the following

A ( -4, 0) ; B (4, 0)

M (2, 0) ; N(8, 0)

and for the radius we get R= 3 and for the center of circumference (c) W (5,0)

Then the equation of the loci for C will be : x^2+y^2-10x +16 =0

It is obvious also that ray MC is the bisector of angle ACB indeed M and N are harmonic conjugates respect to A and B. Having said that and as can be seeing by drawing the angle ACB or the others are somewhat arbitrary except the for bisector property indicated above.

mpdlc


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