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Browse High School Conic Sections, Circles
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    About ellipses.
    Find the center of a circle.
    Is a circle a polygon?
    Volume of a tank.
    Why is a circle 360 degrees?

Inscribed Angle Theorem [01/21/1999]
Can you help me prove that an angle inscribed on the same arc as a central angle is equal to one half that central angle?

Inscribing a Regular Pentagon within a Circle [04/15/1999]
What are the reasons for the steps in inscribing a regular pentagon within a circle with only the help of a compass and a straightedge?

Inscribing a Square within a Half-Circle [04/05/2002]
Is there a way to inscribe a square within a given half-circle?

Intersecting Circles [05/13/1999]
A common tangent touches two intersecting circles at S and T. Show that the line AB bisects the common tangent ST.

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

An Introduction to Parabolas [02/04/1999]
What is a parabola? What are some of its properties? What are some examples of parabolas in real life?

An Introduction to Pi and Its Use in Circle Problems [04/28/2004]
The doctor helps a confused student with circle parts, their relation to each other, and pi.

Involute of a Circle [01/22/1999]
What is the formula for the involute of a circle?

Irregularly Inscribing a Circle ... of What Radius? [12/24/2011]
A student struggles to determine the radius of a circle given the sidelengths of an inscribed irregular polygon. First, Doctor Greenie offers a numerical approach; then, Doctor Floor follows up with the desired analytical method, applying Ptolemy's theorem and the rational root theorem to a key insight: the radius of the circumcircle remains invariant under a re-ordering of the polygon's sidelengths.

Is a Circle a Polygon? [03/07/1999]
A circle is not a polygon.

Lateral Surface of a Cone [02/10/1999]
How do you find the formula for the lateral surface of a right cone?

Latus Rectum [08/08/2002]
I'm trying to find the definition, an explanation, a formula, or anything that will help me to better understand what latus rectum is.

Line and Unit Circle; Pythagorean Triples [04/16/2001]
If (X,Y) is a point in the 1st quadrant on the unit circle and m is the slope of the line passing through (X,Y) and the point (0,-1), how can I express the coordinates (X,Y) in terms of m? Can this be used to generate Pythagorean triples?

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.

Line Tangent to Two Circles [07/01/1999]
How do you construct a line tangent to two different-sized circles?

Locus of the Midpoint of a Chord [03/13/1998]
Show that the locus of the midpoint of the chord is a hyperbola.

Longitude Degrees at the Equator [09/09/1997]
What is the distance in miles between degrees of longitude at the equator?

Mathematics and Intuition [07/10/2001]
Certain "puzzlers" in mathematical recreations defy our sense of experience, leaving you wondering if the answer to a problem can really be true. How to convince the intelligent non-believer?

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?

Minimum Distance from a Point to a Line [7/10/1995]
Find all the values of b such that the minimum distance from the point (2,0) to the line y = 4/3x+b is 5.

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?

Mixtilinear Incircle Proof [12/11/2000]
In triangle ABC, AB = AC. A circle is tangent internally to the circumcircle of triangle ABC and also to sides AB and AC at points P and Q, respectively. How can I prove that the midpoint of the segment PQ is the center of the mixtilinear incircle of triangle ABC?

Mutually Tangent Circles [05/22/2000]
Circles A, B and C are mutually tangent and each is tangent to line DFE at points D, F and E respectively. What is the radius of C as a function of the radii of A and B?

Non-Algebraic Explanation of a Parabola [05/16/2000]
What does a parabola look like? How is it formed?

One More Point Than a Line [07/05/1998]
In terms of 1-1 correspondence, why does a circle have one more point than a line?

The Origin of Conic Sections [10/02/1998]
What are the origins of conic sections?

Origin of Radians [05/27/2002]
Where, exactly, did radians come from?

Overlapping Circles [07/14/1999]
Each of two overlapping circles has a radius of 6 inches. How long is the darkened portion, in inches?

Painting a Merry-Go-Round [06/06/2002]
You have two concentric circles. A chord of the larger, which measures 70 feet, is tangent to the smaller. What is the area of the region between the circles?

Parabola [06/06/2001]
What is the meaning of the word parabola?

Parabola Given Three Points [03/27/2002]
How can I find the equation for a parabola given three points on it?

Parabola: Proving Parallel Rays [05/21/2003]
Given only the geometric definition of a parabola (all points equally distant from focus and directrix), provide a simple geometric proof that all rays emanating from the focus reflect from the parabola parallel to the axis of symmetry.

Parabolas, -b/2a ? [03/13/2002]
I've thought about why the form -b/2a works when trying to graph a parabola, but I just cannot figure it out.

Parabolas in Everyday Life [05/18/2001]
What types of things have been made using parabola shapes?

Parabolas Through Pairs of Points [07/13/2002]
Can infinitely many parabolas pass through any two points in a plane?

Parabolic Equation of an Arch [05/25/1998]
Finding the equation of the parabola in the McDonald's golden arches.

Parabolic Golf Shot Equations [01/24/2002]
Does the ball reach the green?

Pascal's Theorem [10/07/1998]
Can you explain Pascal's Theorem? How does it relate to conic sections?

Perimeter of an Ellipse [06/29/1999]
Is there a formula that gives the exact perimeter of an ellipse?

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