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Perigal Proof of Pythagorean Theorem

Perigal Proof of Pythagorean Theorem
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2025-08-04 19:40
TriangleSolid GeometryGeometric Construction
In collections Satz des Pythagoras - Einführung
This example demonstrates the Perigal proof of the Pythagorean theorem, showcasing a special quadrilateral's properties. The core mathematical concepts involved include the Pythagorean theorem, geometric shapes' similarity and symmetry. The construction uses GeoGebra's Sequence, Point, Angle, and Segment commands. Suitable for middle school mathematics education and basic geometry knowledge.

GeoGebra Commands & Parameters

  1. Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {9, 3, 5, 6, 2, 10, 10, 10, 10, 4}
  2. Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
  3. Textfield(input) → inputBox
  4. Join({Sequence[Segment[Corner[4] + (RasterPixMat * ((k * RasterStepXY), 0)), Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY))], k, 0, RasterMaxX / RasterStepXY], Sequence[Segment[Corner[4] + (RasterPixMat * (0, (k * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], k, 0, RasterMaxY / RasterStepXY]}) → RasterStrecken
  5. Join({Sequence[Text[UnicodeToLetter[65 + k], Midpoint[Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY)), Corner[4] + (RasterPixMat * (((k + 1) * RasterStepXY), RasterMaxY))], false, true, 0, -1], k, 0, RasterMaxX / RasterStepXY - 1], Sequence[Text[("" * k), Midpoint[Corner[4] + (RasterPixMat * (RasterMaxX, ((k - 1) * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], false, true, 1, 0], k, 1, RasterMaxY / RasterStepXY]}) → RasterBeschriftung
  6. Sequence(Sequence[Corner[4] + (RasterPixMat * ((k * RasterStepXY), (m * RasterStepXY))), k, 0, RasterMaxX / RasterStepXY], m, 0, RasterMaxY / RasterStepXY) → RasterPunkte
  7. Sequence(Midpoint[PosA, PosB] + (lengthC / 2 * (cos(k°), sin(k°))), k, 30, 150, 5) → PosListC
  8. Point(PosListC) → PosC
  9. Angle(PosB, PosC, PosA) → rightAngle
  10. Midpoint(PosB, PosC1) → CenterA
  11. Midpoint(PosA, PosC2) → CenterB
  12. Intersect(Line[CenterA, xAxis], randA1(1)) → PosP1, PosP2
  13. Intersect(Line[CenterA, yAxis], randA1(1)) → PosP3, PosP4
  14. If(polyA1(1) > polyB1(1), 1, 2) → unterteilung
  15. Intersect(Line[CenterB, xAxis], randB1(1)) → PosQ1, PosQ2
  16. Intersect(Line[CenterB, yAxis], randB1(1)) → PosQ3, PosQ4
  17. Midpoint(PosA, PosB3) → CenterC
  18. Text("\textsf{\bold{\begin{array}Zieh mich!\end{array}}}", PosC + (0.5, 1.25), false, true, 0, 0) → CapPosC
  19. Segment(Midpoint[Corner[CapPosC, 1], Corner[CapPosC, 2]], Midpoint[Corner[CapPosC, 1], Corner[CapPosC, 2]] + (0.5 * Vector[Midpoint[Corner[CapPosC, 1], Corner[CapPosC, 2]], PosC])) → arrowZiehMich
  20. Segment(PosA, PosB) → segC
  21. Segment(PosB, PosC) → segA
  22. Segment(PosA, PosC) → segB
  23. PolyLine(PosSB, Corner[txt012, 2] + (-0.25, 0), Corner[txt012, 1] + (-0.1, 0), Corner[txt012, 4] + (-0.1, 0.1), Corner[txt012, 3] + (0.1, 0.1), Corner[txt012, 2] + (0.1, 0.35), PosSB) → sb01
  24. Angle(Vertex[segSkizze01(1), 1], Vertex[segSkizze01(1), 3], Vertex[segSkizze01(1), 2]) → AngleSkizze
  25. If(unterteilung ≟ 1, polyA1, polyB1) → poly03
  26. SelectedIndex(dropdown051) → index051
  27. SelectedIndex(dropdown052) → index052
  28. SelectedIndex(dropdown031) → index031
  29. SelectedIndex(dropdown032) → index032
  30. SelectedIndex(dropdown033) → index033
  31. If(step ≟ 3, anim03 ≟ 1, step ≟ 3.5, correctdd031 ∧ correctdd032 ∧ correctdd033, step ≟ 4, anim ≟ 5, step ≟ 5, correctdd051 ∧ correctdd052, true) → clickableBtnNext
  32. If(unterteilung ≟ 2, randB1, randA1) → rand03
  33. If(unterteilung ≟ 2, PosM15, PosM25) → MP
  34. Element({Polygon[PosA, PosB, PosB3, PosA3]}, 1) → polyC
  35. Segment(PosCSquared + (-0.4, -0.2), PosCSquared + (-0.4, -0.2) + (-1.225, 0)) → arrowCSquared

AI Drawing Prompt Reference

Draw a right triangle on the plane using three points A, B, and C, such that the angle at A is a right angle, and connect A–B, A–C, and B–C with straight segment sides. Mark the right angle clearly with an angle indicator. Then construct midpoints on the relevant sides and use them to form a special quadrilateral-like boundary by joining the midpoint-based points and connecting them in a single outline/polyline. Add key labels at the important geometric points to emphasize the Pythagorean proof setup, and include an arrow-style segment pointing toward the central constructed region to guide the viewer’s attention to the relationship between sides and midpoints.
The prompt above was compiled from the original drawing commands and is for reference only.