Search Results for “Tangent”
Sinusoidal Tangent MirrorsLook at a point on a sinusoidal curve (y=sin(x)) with a tangent line. Then, look at a point on the negative of your sinusoidal curve (y=-sin(x)) that is a mirror image of your first point. Then, notice that these mirror image points have mirror image tangent lines.
Euclids Elements - Book 3 - Proposition 14Creating a tangent on a circle given point A that is outside the circle.
Tangent TriangleWe show the areas of the triangle formed by joining 3 points on a parabola, and that formed by the tangents to the parabola at those points.
Parabola Tangent Circumcircle3 tangents of a parabola form a triangle. Its circumcircle passes through the parabola's focus.
circle intersectionsCan you conjecture a formula for the product of the two distances from a point to a circle?
Circles, Tangents and Nonagon DiagonalsYou 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!
Rotating Ellipses with ArcsNow that you know how two ellipses can rotate such that they are tangent to each other and a third larger ellipse (see this app); the next challenge is to figure out how an arc placed on each ellipse can be constrained (with proportional points along the ellipses) such that each arc always covers half of the ellipse and one endpoint is on the tangent point of the two smaller ellipses.
Circles, Tangents, and Heptagon DiagonalsTwo 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!
parabola envelopeWe use a trick to let the trace "open up" as you drag a point. The trick is this: an initial point is given parametric location s*t, create a tangent at this point and its envelope as s varies. Now hide the original point and create another point with parameter t, and make it draggable. Dragging the new point changes the value of t and we see a trace from 0 to t.
Hyperbola Polar LineThe polar line is the locus of the intersections of tangent lines at the ends of chords of teh parabola through a fixed point. Turns out to be conceptually important - not just a curiosity.
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