KNOWPIA
WELCOME TO KNOWPIA

In geometry, the **inverse Pythagorean theorem** is as follows:^{[1]}

Base Pytha- gorean triple |
AC |
BC |
CD |
AB
| |
---|---|---|---|---|---|

(3, 4, 5) | 20 = 4× 5 |
15 = 3× 5 |
12 = 3× 4 |
25 = 5^{2} | |

(5, 12, 13) | 156 = 12×13 |
65 = 5×13 |
60 = 5×12 |
169 = 13^{2} | |

(8, 15, 17) | 255 = 15×17 |
136 = 8×17 |
120 = 8×15 |
289 = 17^{2} | |

(7, 24, 25) | 600 = 24×25 |
175 = 7×25 |
168 = 7×24 |
625 = 25^{2} | |

(20, 21, 29) | 609 = 21×29 |
580 = 20×29 |
420 = 20×21 |
841 = 29^{2} | |

All positive integer primitive inverse-Pythagorean triples having up to three digits, with the hypotenuse for comparison |

- Let
*A*,*B*be the endpoints of the hypotenuse of a right triangle*ABC*. Let*D*be the foot of a perpendicular dropped from*C*, the vertex of the right angle, to the hypotenuse. Then

This theorem should not be confused with proposition 48 in book 1 of Euclid's *Elements*, the converse of the Pythagorean theorem, which states that if the square on one side of a triangle is equal to the sum of the squares on the other two sides then the other two sides contain a right angle.

The area of triangle *ABC* can be expressed in terms of either *AC* and *BC*, or *AB* and *CD*:

given *CD* > 0, *AC* > 0 and *BC* > 0.

Using the Pythagorean theorem,

as above.

The cruciform curve or cross curve is a quartic plane curve given by the equation

where the two parameters determining the shape of the curve, *a* and *b* are each *CD*.

Substituting *x* with *AC* and *y* with *BC* gives

Inverse-Pythagorean triples can be generated using integer parameters *t* and *u* as follows.^{[2]}

If two identical lamps are placed at A and B, the theorem and the inverse-square law imply that the amount of light received at C is the same as when a single lamp is placed at D.

- Pythagorean theorem – Relation between sides of a right triangle