Pythagoras' from Bottema's

Bui Quang Tuan found a way to derive the Pythagorean theorem from an elegant lemma of his. The lemma can be also used to deduce the Pythagorean theorem from that of Bottema. The applet below illustrates the latter derivation.

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Let ΔABC be right at C. Erect right isosceles triangles AA'C and BB'C externally to ΔABC. According to Bottema's theorem, the midpoint M of A'B' is the apex of the isosceles right triangle ABM (and thus is independent of C.) Point C also lies on A'B'. (This is because the angles at C add up to 180°: 45° + 90° + 45° = 180°.)

The configuration is exactly that of Bui Quang Tuan's lemma: AA'||BC and BB'||AC. Which allows one to conclude that:

Area(AA'C) + Area(BB'C) = 2 Area(AMB).

Letting, as usual, a = BC = B'C, b = AC = A'C, and c = AB = 2·OM, we can express the equality of the areas as

b²/2 + a²/2 = 2·c/2·c/2,

which is the Pythagorean identity.

The same configuration admits a different interpretation.

For a right ΔABC, the circumcircle (O) is centered at the midpoint O of the hypotenuse AB and has the radius of c/2. Let M be the midpoint of arc ACB. Form circles A(C) centered at A and B(C) centered at B, both passing through C. Note points A' and B' of intersection of line CM with the two circles.

In circle (O), arc(AM) = arc(BM) = 90°; hence ∠ACA' = ∠BCB' = 45°. This means that triangles AA'C and BB'C are right isosceles at A and B, respectively. We are now in position to apply a theorem of two intersecting circles and then, as before, the area lemma.

Bottema's Theorem

  1. Bottema's Theorem
  2. An Elementary Proof of Bottema's Theorem
  3. Bottema's Theorem - Proof Without Words
  4. On Bottema's Shoulders
  5. On Bottema's Shoulders II
  6. On Bottema's Shoulders with a Ladder
  7. Friendly Kiepert's Perspectors
  8. Bottema Shatters Japan's Seclusion
  9. Rotations in Disguise
  10. Four Hinged Squares
  11. Four Hinged Squares, Solution with Complex Numbers
  12. Pythagoras' from Bottema's
  13. A Degenerate Case of Bottema's Configuration
  14. Properties of Flank Triangles
  15. Analytic Proof of Bottema's Theorem
  16. Yet Another Generalization of Bottema's Theorem
  17. Bottema with a Product of Rotations
  18. Bottema with Similar Triangles
  19. Bottema in Three Rotations
  20. Bottema's Point Sibling

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