This graphical 'proof' of
the Pythagorean Theorem
starts with the
right triangle
below, which has sides of length a, b and c.
It demonstrates that
*a ^{2} + b^{2} = c^{2}*, which is the Pythagorean Theorem.

It is not strictly a *proof*,
since it does not prove every step (for example it does not prove that the empty
squares really are squares). But
it does demonstrate the theorem in an interesting way.

Instructions Click on 'Next' to go through the proof one step at a time, or click on 'Run' to let it run without stopping.

Step 1 | Make 3 copies of the original triangle and arrange the four triangles in a square as shown. The outer square JKLM will remain fixed throughout the rest of the proof. |

Step 2 | Each side of the empty square in the middle has a length of c, and so has an area of
c. ^{2} |

Step 3 | Re-arrange the triangles as shown so that the empty space is now divided into two smaller squares. |

Step 4 | Notice that the top left empty square has each side equal to a, so its area is a.^{2} |

Step 5 | Notice also that the bottom right empty square has each side equal to b,
so its area is b.^{2} |

Step 6 | Done. We have rearranged the triangles inside a constant-size square.
The empty space we started with ( c ) must be equal to the sum of the two empty spaces at the end.
^{2}Therefore a
QED.
^{2}+b^{2} = c^{2} |

- Triangle definition
- Hypotenuse
- Triangle interior angles
- Triangle exterior angles
- Triangle exterior angle theorem
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Pythagorean triples
- Triangle circumcircle
- Triangle incircle
- Triangle medians
- Triangle altitudes
- Midsegment of a triangle
- Triangle inequality
- Side / angle relationship

- Perimeter of a triangle
- Area of a triangle
- Heron's formula
- Area of an equilateral triangle
- Area by the "side angle side" method
- Area of a triangle with fixed perimeter

- Right triangle
- Isosceles triangle
- Scalene triangle
- Equilateral triangle
- Equiangular triangle
- Obtuse triangle
- Acute triangle
- 3-4-5 triangle
- 30-60-90 triangle
- 45-45-90 triangle

- Incenter of a triangle
- Circumcenter of a triangle
- Centroid of a triangle
- Orthocenter of a triangle
- Euler line

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All rights reserved