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Analytic Geometry [08/31/1997]
How do I find the standard equations of the circles that pass through (2,3) and are tangent to both the lines 3x - 4y = -1 and 4x + 3y = 7?

Analytic Proof that Midpoints Form a Circle [03/10/1998]
Analytic proof that midpoints between a point within a circle and its circumference form a circle.

Are Angles Dimensionless? [08/31/2003]
If you look at the dimensions in the equation arc length = r*theta, it appears that angles must be dimensionless. But this can't be right. Or can it?

Area of an Ellipse Cut by a Chord [05/26/2000]
How can you calculate area of the part of an ellipse cut off by a chord, if you know the major and minor axes, and the chord?

Area of a Segment from Arc and Chord Length [11/27/2000]
How do you find the area of a segment of a circle if you know only the arc length and chord length?

Building a Circular Horse Pen [06/16/2002]
My Dad and I are building a round pen for our horse. We have 16 16ft. panels and a 10 ft. gate and a 4ft. gate. (270 ft. total) We want to use a radius and mark the places to dig holes for each post that will support the panels, but we don't know how long the radius should be. Can you help?

Catenary and Parabola Comparison [04/06/2004]
What is the difference between a catenary and a parabola? I don't know the difference in shape. Why is the St. Louis arch a catenary and not a parabola?

Centering Circles [10/05/2002]
Two metal disks need to be centered on each other, but the circle with the larger diameter has the center cut out. How can you center them by knowing the diameters?

Center of Mass of a Semicircle [06/14/1999]
Is there a standard formula I can use to know where the center of mass of a semicircle is?

Changing Angle of a Tank [06/11/2003]
Points A and B represent pressure sensors in fixed positions on the base of a round tank. The chord through CD represents the water level in the tank. Lines a and b are the heights of water registered by each sensor...

Circle Chords [11/4/1994]
Place n distinct points on the circumference of a circle and draw all possible chords through pairs of these points. Assume that no three of these chords pass through the same point. Find and solve the recurrence relation for the number of interior intersection points formed inside the circle.

Circles around a Larger Circle [07/26/2003]
Is there a formula to determine the diameter of several smaller circles outlining the circumference of a larger circle?

Complete This Square: x^2 + y^2 - 6x + 2 = 0 [03/16/2003]
I thought completing the square only involved quadratic functions, such as f(x) = ax^2 + bx + c = 0. This question relates to the equation of a circle...

Completing the Square in a Hyperbola Equation [01/04/2004]
A discussion of how to complete the square and how to apply it to the equation of a hyperbola such as 4(x^2 4x) - 9(y^2 + 6y) = 101.

Computing Ellipse Parameters [9/11/1996]
Given five or more points, I want to find the five parameters of an ellipse (center, axes, orientation).

Concentricity of a Tube [05/09/2003]
I am having trouble finding the concentricity of a tube with a very small inner and outer diameter.

Conic Section of an Elliptical Cone [08/01/2007]
If a flashlight with an elliptical beam is shining on the wall at a slant, is the spot elliptical and, if so, how can the parameters be computed?

Coordinate Geometry of Circles [02/26/2003]
The line with equation y = mx is a tangent to the circle with equation x^2 + y^2 - 6x -6y +17 = 0. Find the possible values of m.

Drawing an Ellipse Using Conjugate Diameters [12/07/2005]
Can you explain how to use Conjugate Diameters to draw an ellipse?

Ellipse in 3D Defined with Parametric Equations [12/21/2003]
Given the lengths of an ellipse's semi-axes, and their directions in 3-space, how do I calculate the amplitudes (Ai) and phases (Bi) of the corresponding parametric equations?

Equation of an Ellipse in 3-Space [07/02/2003]
I am looking for the equation of an ellipse in 3-dimensional space. It can be a parametric formulation (e.g., x(t), y(t), z(t)) or a more canonical form (e.g., the 3D analog to the 2D form ((X*X)/a)+((Y*Y)/ b)=1).

Euclidean Formula for Orthogonal Circles [04/11/2001]
When considering the case when circle C has center at the origin and radius 1, we need to show that the equation of the circle orthogonal to circle C and with center (h,k) is given by: x^2-2hx+y^2-2ky+1=0.

Finding Pairs of Intersecting Chords [07/24/2000]
Consider n chords on a circle, each defined by its end points. Describe an O(n*ln(n)) algorithm for determining the number of pairs of chords that intersect inside the circle.

Finding the Area of Overlapping Circles [11/08/2004]
Given two circles that overlap, find the area of the overlapping region.

Finding the Circumcenter of a Sphere [09/24/2003]
How can I find the circumcenter of a sphere given 3 points on the sphere, along with the radius?

Fitting a Circle to a Given Set of Points [08/12/2005]
Given a set of points how do you calculate the circle of best fit?

Geometry Proof Involving Circle and Triangle [09/26/2005]
Triangle ABC cuts a circle at points E, E', D, D', F amd F'. Prove that if AD, BF and CE are concurrent, than AD', BF' and CE' are also concurrent.

Incenter and Conway's Circle [12/17/2002]
In a triangle, the bisectors of the angles intersect at a point in the interior of the circle. If I use this point as a center to draw a circle, what is the relation of this circle to the triangle?

Inclination of an Ellipse's Major Axis ... in Three Dimensions? [01/09/2011]
An astronomer seeks the least number of points required to uniquely determine an ellipse, given one focus at the origin, in order to subsequently calculate the inclination of its major axis. Doctor George outlines the method, simplifies it, reviews the astronomer's resulting JavaScript code -- then addresses an extension into three dimensions.

Intersecting Circles [4/2/1995]
What is the solution to determine the intersection of two given circles (which don't have the same center and touch in two distinct points)?

Intersection of Circles [01/16/2002]
Given two intersecting circles, find the coordinates of the intersection point(s).

Line Tangent to an Ellipse [03/29/2003]
Find the equation of the tangent to the ellipse x^2 + y^2 = 76 at each of the given points: (8,2),(-7,3),(1,-5). Write your answers in the form y = mx + b.

Maximum Surface Area [07/03/2003]
Within a rectangle x by y, I wish to draw a shape that is no more than x across in any direction, but which has the largest possible surface area within the confines of the rectangle.

Meaning of Value of b in Hyperbola Equation [05/06/2007]
I have been unable to find a justification for why a^2 + b^2 = c^2 in a hyperbola. You can justify a^2 = b^2 + c^2 in an ellipse by looking at special points. Is there a comparable explanation for hyperbolas?

Missile Launch Code [08/03/2003]
What kind of information could you give all 10 people such that if any 3 of them were to get together, they would be able to launch the missiles, but if there were only 2 of them, the information would be insufficient to figure out the code?

Parametric Form of Circle Equation [08/04/2003]
If I know the center and radius of a circle, and three points on the circle, can I find the parametric form of the circle equation in 3D space?

Points within an Ellipse [06/03/2003]
How to generate points that will be within an ellipse.

Probability That Random Chord Exceeds Radius in a Circle [11/20/2004]
If a chord is selected at random on a fixed circle, what is the probability that its length exceeds the radius of the circle?

Proof of Circle Theorem by Vectors [05/03/2001]
Prove, using the vector scalar product, that the angle in a semicircle is always 90 degrees (the hypotenuse being the diameter, and the sides meeting on the perimeter).

Radius of Curvature [07/23/2003]
I have several ellipses whose major and minor diameters I know, but I have no information about their foci.

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