I What is the name of this geometry theorem?

AI Thread Summary
The discussion centers on a geometry theorem related to the conditions under which a triangle can be formed given an angle and two sides, specifically under SSA (side-side-angle) conditions. It explains that if a side length (x) is equal to 10sin(30), exactly one triangle is formed; if x is greater, two triangles can exist; and if x is less, no triangle is formed. This scenario is identified as the ambiguous case of the law of sines. Participants express uncertainty about whether this concept has a formal name, but it is clarified that it relates to the law of sines. The conversation concludes with a focus on the implications of these conditions in triangle formation.
barryj
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In geometry, there is a theorem pertaining to whether given an angle, side, and side gives 0, 1, or 2 triangles. See figure. In the figure, if x = 10sin(30) then there is exactly 1 triangle, if x > 10sin(30) then 2 triangles if x < 10sin(30) then no triangles. I think this has a theorem name or something that I can look up in a tggext book.

img439.jpg
 
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I am not sure I get you. Your drawing is no triangle case, right ? Could you draw "2 triangles" case to confirm I get you properly.
 
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The condition for three sides, A, B, and C to make a triangle is that the sum of any two sides must be greater than the third side. Given an angle and two sides the question is will these form a triangle. There are three conditions depending on how long the "hanging" side is. In my diagram, the hanging side is x the other side is 10, and the angle is 30 deg. If x = 10sin(30) then there will be one right triangle. If x < 10sin(30) then there is no triangle, and if x > 10sin(30) then there will be two triangles. I guess there is not a theorem here, just a problem.
 
Thanks for explanation. Law of cosine
c^2=a^2+b^2-2ab \cos \gamma
Regarding this as quadratic equation of b
b^2-2a \cos \gamma \ b + a^2-c^2 = 0
D/4= (a \cos \gamma)^2 - a^2 + c^2 = ( c- a \sin \gamma )( c + a \sin \gamma)
##a \sin \gamma < c## : b has two real solutions with ##\alpha_1<\frac{\pi}{2}<\alpha_2##
##a \sin \gamma = c## : b has one real solution with ##\alpha=\frac{\pi}{2}##
##a \sin \gamma > c## : b has no real solution

I do not know whether this feature has a specific name or not.
 
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barryj said:
In geometry, there is a theorem pertaining to whether given an angle, side, and side gives 0, 1, or 2 triangles. See figure. In the figure, if x = 10sin(30) then there is exactly 1 triangle, if x > 10sin(30) then 2 triangles if x < 10sin(30) then no triangles. I think this has a theorem name or something that I can look up in a tggext book.
I think it is called the law of Sines, since you are given SSA conditions on the triangle. This case is called the ambiguous case because there could be 0,1, 2 triangles.

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