24
Nov 15

Did Thales teach Pythagoras this?

For any right triangle, the sum of the circles from the lengths of the sides is equal to the circle from the length of the hypotenuse.

New Applet Reveals a Corollary to the Pythagorean Theorem.

Pythagoras said a□ + b□ = c□ yet below we have a◯ + b◯ = c◯

 triangles-circled

The interactive applet is at http://tube.geogebra.org/material/simple/id/2132983

Read on for an explanation of this corollary to Pythagoras' Theorem, without the image or applet.

Draw a perpendicular line from the hypotenuse (h) of a right angle triangle (C) to the intersection of the two lines forming the right angles.

The original right angle triangle (C) now contains two smaller similar right angle triangles, (A) and (B), so it follows A + B = C.

Any regular shape drawn on the hypotenuses of the similar triangles, A, B and C, will also be similar by proportion.

So a circle ◯, triangle △ , square □ and so on drawn on each hypotenuse will also be similar, and given the relation A + B = C, all subsequent similar shapes drawn on the hypotenuses of the similar triangles will also by proportion follow, resulting in Ah◯ + Bh◯ = Ch◯. and Ah△ + Bh△ = Ch△ and Ah□ + Bh□ = Ch□ and so on.

So Pythagoras works for pi(e)s as well!

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