Find points I, K on the sides of a triangle ABC, such that segments CI, IK, KB are equal in length.
Assume that I, K have beein constructed, AC is shorter than AB and the ratio r = CI/CA is known. Define D on AB, so that AD = AC. The homothety fC,r (center at C, modulus r) maps D to a point G, so that IG=IC. Thus, BGIK is a parallelogram and IK=IC=IG, implying that it is a rhombus. Draw a parallel to BG through D intersecting BC at E. E is a constructible point and maps through fC,r on B. Thus, r = BG/ED can be determined from the data. Having the homothety fC,r the construction is obvious.
Problem:
To construct a triangle ABC from the angle A and the sums of its side-lengths a+c and a+b.
Hint: Reduce it to the previous exercise.
See Also
EqualSegments2.html
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