# Euclid's Muse

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# Circles, Tangents, and N-gon Diagonals

A sequence of apps exploring some interesting characteristics of odd-sided regular polygons.

Tags: Circles, tangents, diagonals, n-gons, heptagon, nonagon, hendecagon, tridecagon

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#### Collection Contents:

##### Circles, Tangents, and Heptagon Diagonals
Two circles are centered at intersection points of diagonals of a regular hepatgon. It turns out that circles centered at intersection points in regular polygons (particularly interestingly with polygons of odd numbers of sides) can be tangent to many other diagonals of that polygon. Try resizing the circles by dragging the green points. How many diagonals can each circle be tangent to? Ready for more? Check out the nonagon version!
##### Circles, Tangents and Nonagon Diagonals
You may want to see the heptagon version before attempting this one Every diagonal within a regular nonagon is drawn. Circles are centered at each intersection of diagonals along a vertical axis (these same constructions can be made nine times around the nonagon). Each circle can be tangent to at least 4 diagonals when the circle is at least 2 different sizes. Unnecessary diagonals have been hidden. Drag the green points to resize the circles. Can you find all 13 positions where a circle is tangent to at least 4 diagonals? Hint: sometimes the circle is not entirely contained within the nonagon. Ready for more? Check out the hendecagon version!
##### Hendecagon Diagonals, Circles and Tangents
Before even attempting to understand this app, take a look at the heptagon and nonagon versions. It’s the same situation here, circles centered at intersection points of diagonals within the hendecagon. Drag the green points to resize the circles. Resize the circles so that they are tangent to at least 4 diagonals at the same time (this case is possible in at least two positions for each circle). How many instances can you find on this one? Notice a trend with this and the other versions? Now that you've got this one, check out the final installment, the tridecagon version.
##### Tridecagon Diagonals, Circles and Tangents
You'll want to start out with the heptagon and work your way up. This one's the same as all the others, just with a 13-sided regular polygon. Observe the tangencies to diagonals of circles centered at intersections of diagonals, when the circles are resized (by dragging). This is a smaller version that works well on most monitors (zoom in with two-finger touch). Bigger version here.
##### Tridecagon Diagonals, Circles and Tangents (BIG)
You'll want to start out with the heptagon and work your way up. This one's the same as all the others, just with a 13-sided regular polygon. Observe the tangencies to diagonals of circles centered at intersections of diagonals, when the circles are resized (by dragging). This is a bigger version that works better on very large monitors in fullscreen mode. Smaller version here.