The problem here is to inscribe a particular quadrangle in a triangle. The quadrangle has three sides of the same length, t say, and the fourth of double length, 2t. The shape of these quadrangles is easily controlled by an angle (psi) on their big side. In the first figure above, triangle OMQ is constructible by its angle (psi) at M and MQ=t, MO=2t. Then P is found on the medial line of OQ, so that PQ=PO=t.

The constructed quadrangle of this kind is then transfered by similarity to HLKJ and this, again by similarity, extended to the DEFG, inscribed in the triangle ABC.

I don't enter here in the discussion of the limitations of various kinds controlling the possibility (a) to construct the quadrangle with given t and (psi) and (b) the possibility to inscribe a definite such quadrangle into a triangle with given angles.

The subject is a particular (degenerated equilateral) pentagon inscribed into a triangle. The general case is discussed in Pentadivision.html .

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