Geometriae Dedicata

, Volume 45, Issue 1, pp 115–120

# Triangle in a triangle: On a problem of Steinhaus

• K. A. Post
Article

## Abstract

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.
$$\begin{array}{l} F = {\textstyle{1 \over 4}}(2a^2 b^2 + 2b^2 c^2 + 2c^2 a^2 - a^4 - b^4 - c^4 )^{1/2} \\ F' = {\textstyle{1 \over 4}}(2p^2 q^2 + 2q^2 r^2 + 2r^2 p^2 - p^4 - q^4 - r^4 )^{1/2} . \\ \end{array}$$

## Keywords

Cyclic Permutation Arbitrary Permutation Congruent Copy
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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