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Search Results for “Tangents”

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 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
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 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 Phil Todd
Tangents to Polar Functions
The purple curve has polar equation r=f(θ). A lies on this curve, and B is a point at the intersection of the tangent at A with the line perpendicular to OA. The red curve is the locus of B The grey curve is the curve r=g(θ) rotated by the quantity on the slider. What function g() will rotate to lie on top of the red curve?

Tags: polar-function, tangent, spiral, Archimedes

By Cannon
Death Star vs. Endor
An exciting mix of calculus and the hit series Star Wars, explore a problem the mighty empire could've faced with their final attempt to squash the rebellion.

Tags: Starwars, Derivatives, Normal-lines, Draggable, Fiction, Circles, Ellipses, Tangents

By Phil Todd
Circle isotomic
The circle isotomic is the locus of the reflections of a given point in the tangents to the circle. It is easy to convince yourself that it is also the envelope of the circles whose centers lie on the circle and which pass through the given point. The given point is C.  Try dragging it outside the circle.

Tags: isotomic, curve, envelope, limacon

By admin
Initially you see the cardioid as the caustic curve due to light reflecting in a circle.  (the caustic is the envelope of the reflected rays).  A second view of the cardioid is as the epicycloid formed by a point on the circumference of a disk rolling round a circle of the same radius.  A third view is as the orthotomic (locus of the reflections of the point in the tangents to the curve) of a point on the smaller circle.

Tags: clock

By admin
Circle Pedal
Given a curve and a pole point, the pedal is the locus of the intersections between the tangents to the curve and the perpendiculars through the pole point.  In this case the given curve is a circle, and the pole is the red point.  Try dragging the pole and observe how the clock face changes.  Press the Explain button to see the situation more clearly.

Tags: clock

By Phil Todd
Maximum Area Triangle
A force proportional to the length of each side are applied to the side's midpoint.  This models a constant pressure, and potential energy should be minimized when area is maximized. Press Show to see the tangents to the curve at the points of contact. What do you notice? You can change the definition of the curves, but be careful, or the masses might escape!

Tags: maximum, triangle, area

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