## 1. Triangle Construction-I

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).

## 2. Triangle Construction-II

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.

## 3. Triangle Construction-III

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 .

Bisector.html
Bisector0.html
Bisector1.html
BisectorRectangle.html
Euler.html
PappusTriangleConstruction.html
TriaConstructionAarb.html
TriangleBisectors.html

### References

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