green/blue areas = red area

Pythagoras as a special case:
grüner Bereich = roter Bereich

Im Geometrie, Apollonius's theorem ist ein Satz relating the length of a median of a Dreieck to the lengths of its sides.
It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".

Speziell, in any triangle

wenn

is a median, dann

It is a special case von Stewart's theorem. For an isosceles triangle mit

the median

is perpendicular to

and the theorem reduces to the Pythagorean theorem for triangle

(or triangle

). From the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.

The theorem is named for the ancient Greek mathematician Apollonius of Perga.

Nachweisen[]

Proof of Apollonius's theorem

The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (sehen parallelogram law). The following is an independent proof using the law of cosines.^[1]

Let the triangle have sides

with a median

drawn to side

Lassen

be the length of the segments of

formed by the median, Also

is half of

Let the angles formed between

und

be

und

wo

includes

und

includes

Dann

is the supplement of

und

Das law of cosines zum

und

besagt, dass

Add the first and third equations to obtain

nach Bedarf.

Siehe auch[]

• Formulas involving the medians' lengths – Line segment joining a triangle's vertex to the midpoint of the opposite side

Verweise[]

Externe Links[]

• Apollonius Theorem bei PlanetMath.
• David B. Surowski: Advanced High-School Mathematics. p. 27

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Mathematiker (Zeitleiste)	Anaxagoras Anthemius Archytas Aristaeus the Elder Aristarch Apollonios Archimedes Autolykus Bion Bryson Callippus Carpus Chrysippus Cleomedes Conon Ctesibius Democritus Dicaearchus Diocles Diophantus Dinostratus Dionysodorus Domninus Eratosthenes Eudemus Euklid Eudoxus Eutocius Geminus Heliodorus Heron Hipparch Hippasus Hippias Hippocrates Hypatia Hypsicles Isidore of Miletus Leon Marinus Menaechmus Menelaus Metrodorus Nicomachus Nicomedes Nicoteles Oenopides Pappus Perseus Philolaus Philon Philonides Porphyry Posidonius Proclus Ptolemy Pythagoras Serenus Simplicius Sosigenes Sporus Thales Theaetetus Theano Theodorus Theodosius Theon of Alexandria Theon of Smyrna Thymaridas Xenocrates Zeno of Elea Zeno of Sidon Zenodorus
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