Teorema Stewart

Dalam geometri, teorema Stewart menyatakan hubungan antara panjang sisi-sisi segitiga dan panjang cevian segitiga. Nama teorema Stewart digunakan untuk menghormati matematikawan Skotlandia Matius Stewart yang mempublikasikan teorema ini pada tahun 1746.^[1]

Teorema

Misalkan $a$ , $b$ , dan $c$ panjang sisi-sisi segitiga. Misalkan $d$ panjang cevian pada sisi dengan panjang $a$ . Jika cevian membagi $a$ menjadi dua segmen dengan panjang $m$ dan $n$ dimana $m$ berdekatan dengan $c$ dan $n$ berdekatan dengan $b$ , maka teorema Stewart menyatakan bahwa

b^{2}m+c^{2}n=a(d^{2}+mn).

(persamaan ini juga dapat ditulis

man+dad=bmb+cnc.

, dalam beberapa referensi bahasa inggris, bentuk ini digunakan agar mudah dihafal menjadi sebuah kalimat, misalnya "A man and his dad put a bomb in the sink.")

Teorema Apollonius adalah kasus khusus di mana $d$ adalah garis berat.

Teorema ini juga dapat dinyatakan menggunakan panjang segmen bertanda, yaitu panjang AB bisa positif atau negatif bergantung pada posisi A di sebelah kiri atau di sebelah kanan B pada garis yang sudah di tetapkan. Dalam kasus ini, teorema Stewart menyatakan jika A, B, dan C adalah titik-titik yang berada dalam satu garis, dan P adalah sembarang titik, maka^[2]

PA^{2}\cdot BC+PB^{2}\cdot CA+PC^{2}\cdot AB+BC\cdot CA\cdot AB=0.

Bukti

Teorema Stewart dapat dibuktikan dengan aturan cosinus:^[3]

Misalkan θ sudut antara m dan d dan θ' sudut antara n dan d. θ' adalah suplemen dari θ sehingga cos θ' = −cos θ. dengan menggunakan atura cosinus untuk θ dan θ',

{\begin{aligned}c^{2}&=m^{2}+d^{2}-2dm\cos \theta \\b^{2}&=n^{2}+d^{2}-2dn\cos \theta '\\&=n^{2}+d^{2}+2dn\cos \theta .\,\end{aligned}}

Kalikan persamaan pertama dengan n, persamaan kedua dengan m, dan jumlahkan keduanya untuk mengeliminasi cos θ, diperoleh

{\begin{aligned}&b^{2}m+c^{2}n\\&=nm^{2}+n^{2}m+(m+n)d^{2}\\&=(m+n)(mn+d^{2})\\&=a(mn+d^{2}),\\\end{aligned}}

dan bukti selesai.

alternatif lain, teorema ini dapat dibuktikan dengan menggambar garis tegak lurus dari titik sudut segitiga ke sisi yang berhadapan dan menggunakan teorema Pythagoras untuk menentukan panjang b, c, dan d.

Catatan

^ Stewart, Matthew (1746), Some General Theorems of Considerable Use in the Higher Parts of Mathematics, Edinburgh: Sands, Murray and Cochran "Proposition II"
^ Russell 1905, p. 3
^ Proof of Stewart's Theorem di PlanetMath.

Referensi

Russell, John Wellesley (1905), "Chapter 1 §3: Stewart's Theorem", Pure Geometry, Clarendon Press, OCLC 5259132

Bacaan lebih lanjut

I.S Amarasinghe, Solutions to the Problem 43.3: Stewart's Theorem (A New Proof for the Stewart's Theorem using Ptolemy's Theorem), Mathematical Spectrum, Vol 43(03), pp. 138 – 139, 2011.
Ostermann, Alexander; Wanner, Gerhard (2012), Geometry by Its History, Springer, hlm. 112, ISBN 978-3-642-29162-3

Pranala luar

(Inggris) Weisstein, Eric W. "Stewart's Theorem". MathWorld.
Stewart's Theorem di PlanetMath.

Konten pada artikel ini adalah terjemahan dari aritkel wikepedia bahasa inggris en:Stewart's theorem dengan melakukan beberapa perubahan

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