GeoGebra Tutorial: Golden Ratio / Power of a Point

Introduction

In my previous post, I included this image, which I’d made in GeoGebra. The image satisfies the conditions of the problem: \(AD\) is tangent to \(\odot P\) and \(\overline{BC} \cong \overline{AD}\). In order to create this image, I created a dynamic GeoGebra image where A, B, P and the radius of P can be changed to demonstrate that the ratio \[\frac{BC}{BA} = \frac{AC}{BC} = \phi\] holds true for all values.

In this entry, I will describe one way to create a GeoGebra model that includes both secants for which this relationship is true, with a given tangent and circle.

Steps

1. Start with a new GeoGebra file (I am using 4.0.37.0 on this computer).
2. Create a slider for the radius of the circle. Make sure the Min is positive. You could instead create a circle with a point on its edge, but using a slider means fewer points on the diagram itself.
3. Create a “circle with center and radius”, which is the second option on the drop-down shown. When you place the point (A), you get a pop-up which asks for a radius; type in the name of the slider.
   
4. Create a point outside the circle. This will be where all the tangent and secant lines intersect.
   
5. Create a variable in the Input box. This will be used to make sure that the ratio between the tangent segment and the external secant segment will be \(\phi\).
   
6. Create the tangent lines between B and \(\odot A\). Tangent is the fifth option under the fourth dropdown.
   
7. Create the point of intersection between the lower tangent line and \(\odot A\). The point of intersection option is on the second dropdown.
   
8. Create the tangent segment \(\overline{BC}\). This is on the third dropdown.
   
9. Now you want to make sure that the ratio between the external secant line and the tangent segment is \(\phi\). Do this by creating the circle \(\odot B\) with a radius calculated from \(BC\). Select the “Circle with Center and Radius” option again, then click on point B and the radius pop-up will appear. Since GeoGebra has called that tangent segment d, that’s what I use in the radius box.
   
10. Note that \(\odot B\) will have two, one, or zero points of intersection with \(\odot A\), depending on how close it is. If necessary, move B so that there are two points of intersection. Then create points at each of these.
    
11. Add lines between B and each of these new points.
    
12. Add points of intersection between these lines and the other side of \(\odot A\).
    
13. You now have all the points you need, but things are getting a bit cluttered with unneeded items. Select the arrow (first icon, first option) and turn off “Show Object” for a, b, e, f, and g.
    
14. Create segments BD, DG, BE, and EF.
    
15. Create a text box that shows the relationships between segments and the ratios. If you’re not comfortable with LaTeX, you can also do this with standard text.
    
16. A, B, and radius are free objects and can be moved around. The rest of the points will move accordingly. You can rename any or all objects, and change the rounding, as desired.