In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that the sum of the...

plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262...

proportionality from the angle bisector theorem. Proof: By Stewart's theorem (which is more general than Apollonius's theorem), we have b 2 m + c 2 n = a ( d...

(disambiguation) Apollonius point, a triangle center in plane geometry Apollonius' theorem, an elementary geometry theorem about triangles Circles of Apollonius Problem...

bisector theorem (Euclidean geometry) Anne's theorem (geometry) Apollonius's theorem (plane geometry) Barbier's theorem (geometry) Beck's theorem (incidence...

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle...

into two segments of equal length), the result is known as Apollonius' theorem. The theorem can be proved as an application of the law of cosines. Let...

Euclid is in Apollonius' prefatory letter to the Conics (early 2nd century BC): "The third book of the Conics contains many astonishing theorems that are...

verisimilar to Apollonius. Other than a single other work surviving in Arabic translation, De Rationis Sectione, The rest of Apollonius's works are fragmentary...

Hence by Thales's theorem P {\displaystyle P} lies on the circle which has C D {\displaystyle CD} as a diameter. The Apollonius pursuit problem is one...

famous note about Fermat's Last theorem in his copy of Arithmetica. Descartes, working through the Problem of Apollonius from his edition of Pappus, proved...

\cos ^{2}\!{\frac {A-B}{2}}.} Apollonius' theorem Apollonius of Perga (262–190 BC), geometer and astronomer Apollonius problem Apollonian circles Isodynamic...

Butterfly theorem, Angle bisector theorem, Apollonius' theorem, British flag theorem, Ceva's theorem, Equal incircles theorem, Geometric mean theorem, Heron's...

published a number of translations of major works of Euclid, Archimedes, Apollonius of Perga and others; most are still used today. Heath wrote in the preface...

universe. 200 BC: Apollonius of Perga discovers Apollonius's theorem. 200 BC: Apollonius of Perga assigns equations to curves. 200 BC: Apollonius of Perga develops...

proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number...

equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the...

compass construction. For instance, the problem of Apollonius, closely related to Descartes' theorem, asks for the construction of a circle tangent to...

pair are congruent. The lengths of the medians can be obtained from Apollonius' theorem as: m a = 1 2 2 b 2 + 2 c 2 − a 2 {\displaystyle m_{a}={\frac {1}{2}}{\sqrt...

astronomy as modeled by the celestial sphere. Primarily consisting of theorems which were known at least informally a couple centuries earlier, the Spherics...

Copernicus. This concerns the "Urdi lemma" in particular, an extension of Apollonius' theorem that allowed an equant in an astronomic model to be replaced with...

construction Results Apollonius's theorem Aristarchus's inequality Heron's formula Law of sines Menelaus's theorem Pappus's area theorem Problem II.8 of Arithmetica...

In geometry, Monge's theorem, named after Gaspard Monge, states that for any three circles in a plane, none of which is completely inside one of the others...

Mathematics, Springer, URL: http://encyclopediaofmath.org/index.php?title=Apollonius_theorem&oldid=17516 . Blake, E. M. (1900). "The Ellipsograph of Proclus"....

used to give elegant modern proofs of two classical theorems known to Apollonius. The first theorem is that a closed conic section (i.e. an ellipse) is...

Problem of Apollonius. The idea of the power of a point with respect to a circle can be extended to a sphere . The secants and chords theorems are true...

circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs...

including the case of the parabola (which is lacking in Apollonius's known works). Apollonius's work was translated into Arabic, and much of his work only...

Apollo's belt Apollonius of Perga, Greek mathematician and astronomer – Apollonian circles, Apollonian gasket, Apollonian network, Apollonius' theorem, Problem...

of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained...