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Search Results for “Circles”

By Nick Halsey
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.

Tags: Hendecagon, tangents, circles, diagonals, puzzler

By Nick Halsey
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!

Tags: Heptagon, circles, tangents, diagonals, geometry

By Nick Halsey
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.

Tags: Tridecagon, diagonals, circles, tangents, intersections, puzzler, intricate, confusing, wow

By Nick Halsey
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!

Tags: Nonagons, Circles, Diagonals, Tangents, Puzzler

By L. Van Warren
The Difference of Two Circles Is A Line
This example shows the derivation in wxMaxima and the implementation in Geometry Expressions. The interested user is urged to work through both!

Tags: implicit-equations, circles, lines

By Phil Todd
Tangential Circles
Can you come up with a formula for the radius of the third circle?

Tags: pappus, apollonian-circles

By Nick Halsey
Circle Radius and Proportional Position Puzzler
Five circles are centered proportionally around an ellipse according to functions of the variable r. Each circle’s radius is defined by the same function as its position, of r. Can you figure out the five functions of r which define the five circles? Download the .gx source for the answer. Tip: if you press the ”go” button, r will animate at a constant rate and the ellipse will stay a constant size. If your drag the slider, note that the ellipse is not changing size or position, but the app is zooming in and out in order to display everything within its window most efficiently. Hint: one circle’s radius and position are defined by f(r) = r.

Tags: Circles, radius, proportional-position, puzzler

By Phil Todd
Archimedes Twins
The large circle is tangent to each of the purple circles, which are tangent to each other.  The radius of the large circle is the sum of the radii of the purple circles. The Yellow circles are each tangent to one of the purple circles, to the large circle and to the common tangent of the two purple circles. The yellow circles are called the Archimedes Twins, and have the same radius. Can you work out what this radius is in terms of the radii of the purple circles?

Tags: Archimedes, Twins, circle, tangent

By Natalie Burback
Tangent Lines
Use derivatives to demonstrate the concept of tangent lines with circles.

Tags: Calculus, derivatives, tangent-lines, circles, high-school

By Phil Todd
DG flower
Remember drawing a flower with compasses and coloring it in. Relive the experience electronically.

Tags: Hexagon, circles, flower

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