Triangle in a triangle : on a problem of Steinhaus

K.A. Post

doi:10.1007/BF01667408

Triangle in a triangle : on a problem of Steinhaus

K.A. Post

Mathematics and Computer Science

Research output: Contribution to journal › Article › Academic › peer-review

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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 Max{F(q²+r²-p²), F'(b²+c²-a²)} + Max{F(p²+r²-q²),F'(a²+c²-b²)} = 2Fcr satisfied. In this formula F and F: F=¼(2a²b² + 2b²c² + 2c²a² - a4 - b4 - c4) ¹'² F=¼(2p²q² + 2q²r² + 2r²p² - p4 - q4 - r4) ¹'²

Original language	English
Pages (from-to)	115-120
Journal	Geometriae Dedicata
Volume	45
Issue number	1
DOIs	https://doi.org/10.1007/BF01667408
Publication status	Published - 1993

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title = "Triangle in a triangle : on a problem of Steinhaus",

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 Max{F(q²+r²-p²), F'(b²+c²-a²)} + Max{F(p²+r²-q²),F'(a²+c²-b²)} = 2Fcr satisfied. In this formula F and F: F=¼(2a²b² + 2b²c² + 2c²a² - a4 - b4 - c4) ¹'² F=¼(2p²q² + 2q²r² + 2r²p² - p4 - q4 - r4) ¹'²",

author = "K.A. Post",

year = "1993",

doi = "10.1007/BF01667408",

language = "English",

volume = "45",

pages = "115--120",

journal = "Geometriae Dedicata",

issn = "0046-5755",

publisher = "Springer",

number = "1",

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TY - JOUR

T1 - Triangle in a triangle : on a problem of Steinhaus

AU - Post, K.A.

PY - 1993

Y1 - 1993

N2 - 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 Max{F(q²+r²-p²), F'(b²+c²-a²)} + Max{F(p²+r²-q²),F'(a²+c²-b²)} = 2Fcr satisfied. In this formula F and F: F=¼(2a²b² + 2b²c² + 2c²a² - a4 - b4 - c4) ¹'² F=¼(2p²q² + 2q²r² + 2r²p² - p4 - q4 - r4) ¹'²

AB - 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 Max{F(q²+r²-p²), F'(b²+c²-a²)} + Max{F(p²+r²-q²),F'(a²+c²-b²)} = 2Fcr satisfied. In this formula F and F: F=¼(2a²b² + 2b²c² + 2c²a² - a4 - b4 - c4) ¹'² F=¼(2p²q² + 2q²r² + 2r²p² - p4 - q4 - r4) ¹'²

U2 - 10.1007/BF01667408

DO - 10.1007/BF01667408

M3 - Article

VL - 45

SP - 115

EP - 120

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

IS - 1

ER -