# Euclid's Muse

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# Search Results for “puzzler”

##### Polar Proportional Point and Circle Puzzler 3
A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, u(t), and is located at the current value of t. Adjust t, or animate it by pressing ”go”. What is u(t)? Look at the .gx source for the answer. Hint: if you look closely, u(t) can be seen in dark gold. It is a particular ”polar flower”.

Tags: Polar-function, proportional-point, circle, radius, flower, puzzler

##### 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.

##### Polar Proportional Point and Circle Puzzler 2
A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, s(t), and is located at the current value of t. Adjust t, or animate it by pressing ”go”. What is s(t)? Look at the .gx source for the answer.   Hint: if you look closely, s(t) can be seen in dark purple. It is a particular ”polar flower”.

Tags: Polar-function, proportional-point, circle, radius, flower, puzzler

##### Polar Proportional Point and Circle Puzzler
A circle is centered at the origin and its radius is defined by the distance between the origin and a point, P. P is defined by a polar function, r(t), and is located at the current value of t. Adjust t, or animate it by pressing ”go”. What is r(t)? Look at the .gx source for the answer. Hint: if you look closely, r(t) can be seen in dark blue. It is a particular ”polar flower”.

Tags: Polar-function, proportional-point, circle, radius, flower, puzzler

##### Epic Circle Trace 3
Four points are located proportionally around a circle, according to four different functions of t. A figure connecting the four points is traced through t. What are the four functions? Look at the .gx source for the answer. Tip: press "go" to animate t at a constant rate from 0 to ∏ and back, looped.

Tags: Proportional-point, function, circle, puzzler, trace, ellipse, sometimes-quadrilateral

##### Epic Circle Trace 2
A triangle, defined by three points that are located proportionally around a circle by functions of t, is traced as t varies from 0 to 2Π. What are the functions of t, f(t), g(t), and h(t), that define the points D, E, and F, respectively? Hint: one of the functions is _(t) = t.

Tags: Trace, puzzler, circle, triangle, proportional-point, functions

##### An Arbelos Theorem
The large circle in the diagram has radius 1.  Circles centered at A and B are tangent and each tangent to the large circle.  (The shape in between the circles is called an arbelos after the Greek for a cobbler's knife, which it apparently resembles.) BC is tangent to circle A and AD tangent to circle B. EF and GH are perpendicular to AB. Can you show that they are equal? Can you find an expression for their length in terms of the radius of circle A? Are they the same size as Archimedes "twins"

Tags: archimedes, arbelos, puzzler

##### Epic Circle Trace
A line passes intersects a circle at two points. Each point is located proportionally around the circle in terms of a given function of t. The path of the line’s movement is traced as t varies. Try changing/animating t. Can you figure out how each point is constrained, in terms of t? Look at the gx source file for the answer. Hint: look at the period of the movement, and how it changes as t changes.

Tags: Trace, puzzler, circle, proportional-points, functions

##### Double Transforming Pentagons
Two pentagons are nested within a third, regular pentagon (ABCDE). They are each defined by a point on each side of ABCDE; these points are proportional along their sides at functions of t. Determine the function of t that represents the proportionality of each point. Notice that both pentagons are congruent to ABCDE when t = 0 and t = 1.

Tags: Pentagon, Proportional, Proportionality, N-gon, Puzzler

##### 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