In Euclidean geometry, the Apollonius point is a triangle center designated as X(181) in Clark Kimberling's Encyclopedia of Triangle Centers (ETC). It is defined as the point of concurrence of the three line segments joining each vertex of the triangle to the points of tangency formed by the opposing excircle and a larger circle that is tangent to all three excircles.
In the literature, the term "Apollonius points" has also been used to refer to the isodynamic points of a triangle.[1] This usage could also be justified on the ground that the isodynamic points are related to the three Apollonian circles associated with a triangle.
The solution of the Apollonius problem has been known for centuries. But the Apollonius point was first noted in 1987.[2][3]
^Katarzyna Wilczek (2010). "The harmonic center of a trilateral and the Apollonius point of a triangle". Journal of Mathematics and Applications. 32: 95–101.
^Kimberling, Clark. "Apollonius Point". Archived from the original on 10 May 2012. Retrieved 16 May 2012.
solution of the Apollonius problem has been known for centuries. But the Apolloniuspoint was first noted in 1987. The Apolloniuspoint of a triangle is...
information on Apollonius remains. The 6th century Greek commentator Eutocius of Ascalon, writing on Apollonius' Conics, states: Apollonius, the geometrician...
plausible reconstruction of Apollonius' method. The method of van Roomen was simplified by Isaac Newton, who showed that Apollonius' problem is equivalent...
up Apollonius in Wiktionary, the free dictionary. Apollonius (Ancient Greek: Απολλώνιος) is a masculine given name which may refer to: Apollonius of Athens...
The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are...
Apollonius of Rhodes (Ancient Greek: Ἀπολλώνιος Ῥόδιος Apollṓnios Rhódios; Latin: Apollonius Rhodius; fl. first half of 3rd century BC) was an ancient...
Apollonius of Alexandria may refer to: Apollonius of Alexandria, winner of the Stadion race of the 218th Olympiad in AD 93 Apollonius of Alexandria, philosopher...
point is also the Fermat point. The Apolloniuspoint is the point of concurrence of three lines, each of which connects a point of tangency of the circle...
{\bigl |}[A,B;C,P]{\bigr |}=1.} Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A, B; C, P] is on the unit...
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example...
Apollonius of Tyre is the hero of a short ancient novel, popular in the Middle Ages. Existing in numerous forms in many languages, all are thought to derive...
hyperbola, but on the right branch. See also Problem of Apollonius. The idea of the power of a point with respect to a circle can be extended to a sphere...
of Apollonius, 95-6 R. F. Glei, Outlines of Apollonian Scholarship 1955–1999, p. 13-15 M. Asper, Apollonius on Poetry, p. 186 M. Asper, Apollonius on...
Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians...
ephaptoménē) to a circle in book III of the Elements (c. 300 BC). In Apollonius' work Conics (c. 225 BC) he defines a tangent as being a line such that...
thread sizes are often denoted in this way. The symbol has a Unicode code point at U+2300 ⌀ DIAMETER SIGN, in the Miscellaneous Technical set, and should...
novelistic biography Life of Apollonius of Tyana. It purports to give a full account of the capture of "Lamia of Corinth" by Apollonius, as the general populace...
of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent...
internally tangent to each of the excircles and is thus an Apollonius circle. The radius of this Apollonius circle is r 2 + s 2 4 r {\displaystyle {\tfrac {r^{2}+s^{2}}{4r}}}...
around 700 BCE, but is best known from Euripides's tragedy Medea and Apollonius of Rhodes's epic Argonautica. As a daughter of King Aeëtes she is a mythical...
Battle of the Ascent of Lebonah (Hebrew: קרב מעלה לבונה) or Battle with Apollonius (Hebrew: קרב אפולוניוס) was the first battle fought between the Maccabees...
intersection of the three lines, and the three exscribed circles. A general Apollonius problem can be transformed into the simpler problem of circle tangent...
forms right angles (angles that are 90 degrees or π/2 radians wide) at the point of intersection called a foot. The condition of perpendicularity may be...
the circles of Apollonius, each isodynamic point is the intersection points of another triple of circles. The first isodynamic point is the intersection...
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