Here is the previous post concerning the Golden Ratio. For a collection of all the posts concerning the Golden Ratio, click #GoldenRatio below.
Since we have already talked about several Golden figures, we are going to delve more deeply into where exactly we can find the golden ratio in other figures. Today we will talk about the ubiquitous triangle. Everyone has learned that this is the simplest of all integral right triangles. But interestingly enough, it also contains the golden ratio. Here is the construction:
Above is the triangle with , , and . Let be the foot of the angle bisector from . Draw a circle centered at with radius . Then extend until it hits circle again at . Let be the other point of intersection of these two curves. Then
Problem 1) Can you prove the above proposition?
Problem 2) Extra: In the figure above, there is another point . Prove that the circle intersects the line segment at only one point . Also, why might it be called ?
Here is the next post.