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
- Start with a new GeoGebra Classic 5 file (I am using 5.0.783 on this computer).
- 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.
- 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.
- Create a point outside the circle. This will be where all the tangent and secant lines intersect.
- Create the tangent lines between B and \(\odot A\). Tangent is the fifth option under the fourth dropdown.
- Create the point of intersection between the lower tangent line and \(\odot A\). The point of intersection option is on the second dropdown.
- Create the tangent segment \(\overline{BC}\). This is on the third dropdown.
- Place a point on \(\odot A\) and then create a secant line to B.
- Place a point at the other intersection of \(\odot A\) and the line \(BD\).
- You want segment \(DE\) to be the same length as \(BC\), so move point B around until the \(\odot D\) intersects line \(DE\) at E.
- Place segments on \(DE\), \(EB\), and \(DB\). At this point, your diagram might look like this:
- In the input box, type the letters corresponding to each segment as a ratio. In the example, this would be j/k and l/j. These should both be close to 1.62, which is (approximately) the Golden Ratio.
- You can now drag points B and D around, as well as adjusting the slider, to see that whenever E is on \(\odot D\), those two ratios are about 1.62.
- For student demonstrations, turn off the extraneous details.
