Heron Triangles: An Incenter Perspective

Sastry, K. R. S.

doi:10.1080/0025570x.2000.11996881

Cited by 3 publications

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“…To honour Hoppe we call a triangle Hoppe triangle if the sides are in arithmetic progression. At this point we mention that there is a vast literature on Heron triangles and the reader is invited to refer to [3,4,6,7] to get a glimpse of the many interesting Heron problems. Proof .…”

Section: The Analogiesmentioning

confidence: 99%

Analogies are interesting!

Sastry¹

2004

Elemente der Mathematik

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obtained his Bachelor's and Master's degrees in mathematics from the University of Mysore in India. He taught mathematics, initially in the State of Karnataka in India, later on in Ethiopia. After his retirement in 1994 he returned to the city of Bangalore in India, where he continues to work on mathematical problems.Let an equilateral triangle ABC be inscribed in a circle. Let P be any point on the minor BC (arc BC) as in Fig. 1. Then it is well-known that AP = BP + PC. (This can be easily proved using Ptolemy's theorem, trigonometry or by other means.) Is not this property pretty? Obviously, we do not expect a non equilateral triangle to exhibit such a property. However, in a given triangle ABC it seems reasonable to expect the existence of at least one point P with the above property. Surprisingly, a non equilateral isosceles triangle does not possess such a point. Our discussion examines the reason and considers interesting special cases. Next, we look at the right angled triangle ABC in which ∠BAC = π/2. If AD is drawn perpendicular to BC (shown in Fig. 2), then it is well-known that AD 2 = BD · DC. In fact, if D is the mid point of BC of the same ∆ABC, then also it trivially follows that AD 2 = BD · D C because D is the center of the circle ABC. The second part of our discussion focuses on the characterization of ∆ABC in which at least one point D exists on BC with the property just mentioned. Before we begin the discussion, let us recall the necessary background euclidean geometry results.. Analogien spielen bekanntlich eine wichtige Rolle beim Entdecken neuer Erkenntnisse in der Mathematik. An diese Tatsache knüpft der Autor in der nachfolgenden Arbeit an. Ausgehend von dem bekannten Ergebnis, dass für einen Punkt P auf dem Umkreis eines gleichseitigen Dreiecks ABC zwischen B und C die Beziehung AP = BP+PC besteht, wird in Umkehrung dazu nach notwendigen und hinreichenden Bedingungen an die Seiten eines beliebigen Dreiecks ABC gesucht, so dass ein Punkt P auf dem Umkreis des gegebenen Dreiecks zwischen B und C existiert, der der obigen Gleichung genügt. Ausgangspunkt für das zweite Beispiel ist der Höhensatz im rechtwinkligen Dreieck. Verallgemeinernd dazu wird nach notwendigen und hinreichenden Bedingungen an die Seiten eines beliebigen Dreiecks ABC gefragt, so dass ein Punkt D auf der Seite BC existiert, der die Beziehung AD 2 = BD · DC erfüllt. 30 Elem. Math. 59 (2004)

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Section: The Analogiesmentioning

confidence: 99%

Analogies are interesting!

Sastry¹

2004

Elemente der Mathematik

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“…Other authors have worked with the elliptic curves that appear from this problem [10,11,19]. More generally, Heron triangles (triangles with rational side lengths and rational area) have been extensively studied by various authors [2,4,14,16,18,21,24,28,33]. More general cevians were studied in [6,22].…”

Section: Introductionmentioning

confidence: 99%