Cyclic Sums for Polygons

Shephard, G. C.

doi:10.1080/0025570x.1999.11996714

“…(vi) The idea to use weights attached to the ratios originated with Shephard [16]; he kindly sent a preprint of this paper to one of us. For polygons in the plane Shephard establishes in [16] a restricted version of part (i) of our Theorem 2. Associating weights to ratios was also used in [7], for ratio-sums of a slightly different kind.…”

Section: Ratio-sums For Simplicesmentioning

confidence: 99%

Euler's Ratio-Sum Theorem and Generalizations

Grünbaum

¹

,

Klamkin

²

2006

Mathematics Magazine

3

0

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Over the centuries, many papers have been written about relations among different parts of a triangle. Well-known mathematicians as well as others have contributed to these discoveries. The main aim of this note is to show how asking the right questions can lead to new facts and to far-reaching generalizations that retain an elementary nature and would have been understandable to mathematicians of ages past.Our starting point is one of the results of Euler's paper [5], which shows thatWe use the notation indicated in Figure 1, with Q an arbitrary point in the plane of the arbitrary triangle A 1 A 2 A 3 , and B i the intersection point of the cevian line Q i Q with the side opposite A i . Here and throughout, the only restriction is that all the points are well-defined and all the lengths appearing in the denominators are not zero. The lengths are understood as signed lengths; since only ratios of collinear segments are considered, the positive 1 Professor Klamkin passed away in the summer of 2004. As a friend and a mathematician he will be sorely missed by many of us. Professor Klamkin was still able to see the referees' comments on our paper, and approve the proposed final version of it. A variety of unfortunate circumstances delayed the sending of that version to the Editor. But this had the silver lining contained in part (vii) of the last section, added September 15, 2005. BG

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“…For polygons in the plane Shephard establishes in [16] a restricted version of part (i) of our Theorem 2. Associating weights to ratios was also used in [7], for ratio-sums of a slightly different kind.…”

Section: Ratio-sums For Simplicesmentioning

confidence: 99%

“…(vi) The idea to use weights attached to the ratios originated with Shephard [16]; he kindly sent a preprint of this paper to one of us. For polygons in the plane Shephard establishes in [16] a restricted version of part (i) of our Theorem 2.…”

Section: Ratio-sums For Simplicesmentioning

confidence: 99%

Euler's Ratio-Sum Theorem and Generalizations

Grünbaum¹,

Klamkin²

2006

Mathematics Magazine

View full text Add to dashboard Cite

Over the centuries, many papers have been written about relations among different parts of a triangle. Well-known mathematicians as well as others have contributed to these discoveries. The main aim of this note is to show how asking the right questions can lead to new facts and to far-reaching generalizations that retain an elementary nature and would have been understandable to mathematicians of ages past.Our starting point is one of the results of Euler's paper [5], which shows thatWe use the notation indicated in Figure 1, with Q an arbitrary point in the plane of the arbitrary triangle A 1 A 2 A 3 , and B i the intersection point of the cevian line Q i Q with the side opposite A i . Here and throughout, the only restriction is that all the points are well-defined and all the lengths appearing in the denominators are not zero. The lengths are understood as signed lengths; since only ratios of collinear segments are considered, the positive 1 Professor Klamkin passed away in the summer of 2004. As a friend and a mathematician he will be sorely missed by many of us. Professor Klamkin was still able to see the referees' comments on our paper, and approve the proposed final version of it. A variety of unfortunate circumstances delayed the sending of that version to the Editor. But this had the silver lining contained in part (vii) of the last section, added September 15, 2005. BG

show abstract