The problem is to construct triangle ABC, knowing the lengths. i) of side a=|BC|,
ii) of altitude ha = |AD|,
iii) and of bisector ba=|AE|.
The clue is to consider the symmetric B' of B on the parallel to side BC. Triangle B'BC is
constructible from the data. The same is true for triangle ADE. Hence angle(DAE) = B-C (see
Bisector.html ) is constructible from the data. Besides, for F on the extension of AC: angle(FAB') = angle(BFA)-angle(FB'A) =
(pi/2-C)-(pi/2-B) = B-C. Thus A is intersection point of the medial line of BB' and the arc on B'C viewing it under the
angle w=pi-(B-C).
To construct triangle ABC, knowing the lengths. i) of side a=|BC|,
ii) of median ma = |AD|,
iii) and of bisector ba=|AE|.
Use the formulas expressing the lengths as functions of the sides (see Stewart.html ).
To solve the system for b, c, set b+c=x and bc=y2. Then solve the first w.r. to y and replace
to the second. This leads to the biquadratic equation, whose solutions can be constructed
with straightedge and compasses:
Solution proposed by Fursenko in his remarkable exposition of triangle constructions, p. 23.
To construct triangle ABC, knowing the measures. i) of angle A,
ii) of median ma = |AD|,
iii) and of bisector ba=|AE|.
Solution after G. Velissarios (AMM 1988, p. 458). Assume the triangle constructed and
take points. B' : symmetric ot B with respect to A, M middle of BC, D : trace on BC of bisector of A. E : trace on CB' of external bisector of A. By the basic bisector relation ( Bisector0.html ). DB/DC = AB/AC = AB'/AC = EB'/EC, hence DE is parallel to BB'. It follows that the right-angled triangle DAE is constructible since |AD|=ba and
its angle at D is A/2. Thus it suffices to construct triangle CAB', for which are known. i) the angle at A, ii) the bisector |AE| and iii) the side |CB'| = 2ma. This kind of construction is discussed in PappusTriangleConstruction.html .
E3134 problem, American Mathematical Monthly 1988, p. 458
Fursenko F. B. Lexicographical account of constructional problems of triangle geometry problems Mathematics in school, 1937, no. 5 p. 23, Moscow
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