Let's think Greek! Surprising geometric solution of quadratic equations

András Ringler
University of Szeged, Institute of Medical and Biophysics
ringler@comser.szote.u-szeged.hu

Let's think Greek!
Surprising geometric solution of quadratic equations

The lecturer will show surprising geometric solutions of quadratic equations, and will show how to construct a regular pentagon from a square or from a triangle; it means, how to construct distances being in golden ratio.

Keywords: geometric solution of quadratic equations, regular pentagon, the secret symbol of Pythagoreans (the pentagramma mirificum).

If an ABCD square, with a side length a is given (a ş AB = BC = CD =DA), then around the middle point of the AB side one has to draw a circle going through the points C and D. This circle and the straight line containing the points A and B cross each other in the points E and F. The points E, F and D being on the circle form a Thales-triangle. The height of the EFD triangle - belonging to the EF hypotenuse - the AD ş a distance is the geometrical mean of the EA ş b and the AF ş a+b distances:
a2 = b× (a+b).
The a2 = b× (a+b) equation is equivalent with the b : a = a : (a+b) equation, which represents the golden ratio, known from the secret symbol of the Pythagoreans. If at the beginning of the construction the b distance is unknown, then by the drawing the above mentined circle, the distance x = b is one solution of a so called hyperbolic symptoma (equation), given in the form
a2 = x× (a+x) = a× x+x2.

It is interesting that the distances a and x = b are in golden ratio, so with these two distances the “pentagramma mirificum”, the secret symbol of the Pythagoreans can be constructed, step by step, with compasses and a ruler.

If in the a2 = a× x+x2 equation the a distance were the unknown (a = y) and the x = b distance were the known one, then one has to solve the y2 = y× b+b2 Ţ b× y - y2 = - b2 elliptical symptoma (equation).

To solve (such kind of or other type of) quadratic equations with compasses and a ruler could be interesting to students and to professors too. The link
www.mozaik.info.hu/homepage/mozaportal/matematika.php
shows an animation at the 4 different symptomas (equations). The program can be seen online, or you can make a free copy to use it later in teaching of mathematics:

till to day the number of visitors is over 21000,
and the number of downloadings is over 9400.

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