# Garfield’s proof of Pythagorean theorem

James Garfield, the ${20}^{\mathrm{th}}$ president of the United States, gave the following proof of the Pythagorean Theorem^{} in 1876. Consider the following trapezoid^{} (note that this picture is half of the diagram used in Bhaskara’s proof of the Pythagorean theorem (http://planetmath.org/ProofOfPythagoreasTheorem)).

Recall that the area of a trapezoid with two parallel^{} sides (in this case, the left and right sides) ${s}_{1}$ and ${s}_{2}$ and height $h$ is

$$h\frac{{s}_{1}+{s}_{2}}{2}$$ |

So the area of the trapezoid above is

$$(a+b)\frac{a+b}{2}=\frac{{(a+b)}^{2}}{2}$$ |

The area of the yellow triangle (and that of the blue triangle) is

$$\frac{ab}{2}$$ |

while the area of the red triangle (also a right triangle) is

$$\frac{{c}^{2}}{2}$$ |

The two areas must be equal, so

$\frac{{(a+b)}^{2}}{2}$ | $=2{\displaystyle \frac{ab}{2}}+{\displaystyle \frac{{c}^{2}}{2}}$ | ||

$\frac{{a}^{2}+2ab+{b}^{2}}{2}$ | $=ab+{\displaystyle \frac{{c}^{2}}{2}}$ | ||

${a}^{2}+2ab+{b}^{2}$ | $=2ab+{c}^{2}$ | ||

${a}^{2}+{b}^{2}$ | $={c}^{2}$ |

Title | Garfield’s proof of Pythagorean theorem^{} |
---|---|

Canonical name | GarfieldsProofOfPythagoreanTheorem |

Date of creation | 2013-03-22 17:09:33 |

Last modified on | 2013-03-22 17:09:33 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 10 |

Author | rm50 (10146) |

Entry type | Proof |

Classification | msc 51-00 |