Euclid's Muse

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Search Results for “tangent”

By Phil Todd
Maximum Area Triangle
The app uses a physical model to find the largest area triangle whose vertices lie on a smooth closed curve (defined by the red control points). The triangle defined by the tangent lines at the vertices is displayed, along with its medians. What do you notice about the maximum?

Tags: maximum-area

By Nick Halsey
Rotating Ellipses
Two ellipses rotate within a third such that the two smaller ellipses are always tangent to each other and the larger ellipse. Adjusting the variable X demonstrates this rotation. All three ellipses are constrained only by their implicit equations, and the moving ellipses' equations include functions of X that allow them to rotate. Can you figure out what the equations are (the other constants in the equations are a, b, h, and k)?

Tags: Ellipses, conics, puzzler

By Phil Todd
Circle in a trapezoid
When is it possible to draw a circle tangent to all four sides of a trapezoid?  

Tags: trapezoid, inscribed-circle

By Nick Halsey
Basic Derivatives
Drag the point to see how the slope of the line relates to the x value of the point at which it’s tangent to the function. Can you figure out what the function is, based on the values of the x and y coordinates? The slope of the line can also be represented in terms of x; can you figure out what this representation is? This representation is the derivative of the entire function, not just at a single point. This is called the derivative of the function, and can be notated by, for example, the derivative of F(x) = F’(x), although there are many other notations as well.

Tags: Calculus, Derivatives, Functions

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 Phil Todd
Parabola Property
The triangle containing the normal and tangent at a point on a parabola sits inside a semicircle centered at the parabola's focus.


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