A fun interactive proof you may like is at http://tube.geogebra.org/m/1544415

In the Geometric Progression A : B : C where A = 1 the product of the extremes A and C equals B^2.

For example, in the Geometric Progression 1, 5, 25 we get 1 × 25 = 5 × 5.

Yet how can we prove the validity of the progression 1, -5, 25? It's not very well known in the mathematics community that proportion is preserved across sign so I have made a fun interactive applet that proves the existence of 2 roots for all positive Reals.

In 300 BCE the Greek mathematician Euclid of Alexandria may not have been aware proportion is preserved across direction of magnitudes. René Descartes was also unaware in 1637 when he published La Géométrie which had the the top half of this diagram copied from Euclid's Book VI Proposition 13.

Fast forward to Isaac Newton and Newton makes it known a line in an opposite direction to an 'affirmative' line may be considered 'negative'. Newton was the first to popularise the depiction of curves spanning the 4 quadrants of the Cartesian Plane.

See also: http://aleph0.clarku.edu/~djoyce/elements/bookVI/propVI13.html

You may also enjoy my applets on the topic of ** Proportional Co-Variation** at

http://tube.geogebra.org/book/title/id/1410639

Best wishes,

Jonathan