# Triangle in a triangle: On a problem of Steinhaus

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## Abstract

A necessary and sufficient condition on the sides is satisfied. In this formula

*p, q, r*of a triangle*PQR*and the sides*a, b, c*of a triangle*ABC*in order that*ABC*contains a congruent copy of*PQR*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$$\begin{array}{l} Max\{ F(q^2 + r^2 - p^2 ), F'(b^2 + c^2 - a^2 )\} \\ + Max\{ F(p^2 + r^2 - q^2 ), F'(a^2 + c^2 - b^2 )\} \le 2Fcr \\ \end{array}$$

*F*and*F*′ 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.

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## References

- 1.Steinhaus, H.,
*One Hundred Problems in Elementary Mathematics*, Pergamon, Oxford, 1964, p. 98.Google Scholar - 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