Construct a triangle ABC from its angle A, the opposite side-length a=BC and the radius rb of the excircle on angle B.
The solution is indicated by the figure. The triangles AEF and BCD are constructible in this order from the data. First AEF is a right angled triangle with EF = rb and the angle at A equal to (π-A)/2. From the constructed triangle, side AF gives the length (s-c), where s = (a+b+c)/2 is the half-perimeter of the triangle. But s-c = (a+b+c)/2 - c = (a+b-c)/2 = (a/2) + (b-c)/2. From this we determine the length (b-c) = DC. Knowing also BC = a and the angle at D, we can construct BDC. From this the construction of ABC is obvious.
Fursenko F. B. Lexicographical account of constructional problems of triangle geometry problems Mathematics in school, 1937, no. 5 pp. 4-30, no. 6 pp. 21-45, Moscow
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Triangle construction (A, a, rb) ![[0_0]](TriaConstructionAarb_files/TriaConstructionAarb_0_0_0.png)