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Trigonometry/For Enthusiasts/A Generalization of Pythagoras' Theorem

Arbitrary triangleEdit

 
Generalization of Pythagoras' theorem by Tâbit ibn Qorra.[1] Lower panel: reflection of triangle ABD (top) to form triangle DBA, similar to triangle ABC (top).

At any selected angle of a general triangle of sides a, b, c, inscribe an isosceles triangle such that the equal angles at its base θ are the same as the selected angle. Suppose the selected angle θ is opposite the side labeled c. Inscribing the isosceles triangle forms triangle ABD with angle θ opposite side a and with side r along c. A second triangle is formed with angle θ opposite side b and a side with length s along c, as shown in the figure. Tâbit ibn Qorra[2] stated that the sides of the three triangles were related as:[3][4]

 

As the angle θ approaches π/2, the base of the isosceles triangle narrows, and lengths r and s overlap less and less. When θ = π/2, ADB becomes a right triangle, r + s = c, and the original Pythagoras' theorem is regained.

One proof observes that triangle ABC has the same angles as triangle ABD, but in opposite order. (The two triangles share the angle at vertex B, both contain the angle θ, and so also have the same third angle by the triangle postulate.) Consequently, ABC is similar to the reflection of ABD, the triangle DBA in the lower panel. Taking the ratio of sides opposite and adjacent to θ,

 

Likewise, for the reflection of the other triangle,

 

Clearing fractions and adding these two relations:

 

the required result.

CreditsEdit

  • Howard Whitley Eves (1983). "§4.8:...generalization of Pythagorean theorem". Great moments in mathematics (before 1650). Mathematical Association of America. p. 41. ISBN 0883853108. http://books.google.com/books?id=9_w5jDPTvCQC&pg=PA41. 
  • Tâbit ibn Qorra (full name Thābit ibn Qurra ibn Marwan Al-Ṣābiʾ al-Ḥarrānī) ( 826–901 AD) was a physician living in Baghdad who wrote extensively on Euclid's Elements and other mathematical subjects.
  • Aydin Sayili (Mar. 1960). "Thâbit ibn Qurra's Generalization of the Pythagorean Theorem". Isis 51 (1): 35–37. doi:10.1086/348837. http://www.jstor.org/stable/227603. 
  • Judith D. Sally, Paul Sally (2007-12-21). "Exercise 2.10 (ii)". Cited work. p. 62. ISBN 0821844032. http://books.google.com/books?id=nHxBw-WlECUC&pg=PA62.