Geometriae Dedicata

, Volume 45, Issue 1, pp 115–120 | Cite as

Triangle in a triangle: On a problem of Steinhaus

  • K. A. Post


A necessary and sufficient condition on the sidesp, q, r of a trianglePQR and the sidesa, b, c of a triangleABC in order thatABC contains a congruent copy ofPQR is the following: At least one of the 18 inequalities obtained by cyclic permutation of {a, b, c} and arbitrary permutation of {itp, q, r} in the formula
$$\begin{array}{l} Max\{ F(q^2 + r^2 - p^2 ), F'(b^2 + c^2 - a^2 )\} \\ + Max\{ F(p^2 + r^2 - q^2 ), F'(a^2 + c^2 - b^2 )\} \le 2Fcr \\ \end{array}$$
is satisfied. In this formulaF andF′ denote the surface areas of the triangles, i.e.
$$\begin{array}{l} F = {\textstyle{1 \over 4}}(2a^2 b^2 + 2b^2 c^2 + 2c^2 a^2 - a^4 - b^4 - c^4 )^{1/2} \\ F' = {\textstyle{1 \over 4}}(2p^2 q^2 + 2q^2 r^2 + 2r^2 p^2 - p^4 - q^4 - r^4 )^{1/2} . \\ \end{array}$$


Cyclic Permutation Arbitrary Permutation Congruent Copy 
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  1. 1.
    Steinhaus, H.,One Hundred Problems in Elementary Mathematics, Pergamon, Oxford, 1964, p. 98.Google Scholar
  2. 2.
    Croft, H. T., Falconer, K. J. and Guy, R. K.,Unsolved Problems in Geometry, Springer, New York, 1990, p. 50.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • K. A. Post
    • 1
  1. 1.Department of Mathematics and CSTechnological UniversityEindhovenThe Netherlands

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