Proving a quadrilateral is cyclic and finding the radius of the circle

In summary, the conversation discusses applying a characterization of a cyclic quadrilateral to solve a problem. The speaker suggests constructing triangle OPQ and using the fact that the perpendicular bisectors of a cyclic quadrilateral are concurrent to find the center of the circumcircle. The speaker also mentions that the hypothesis about the circles being concentric may be relevant but is unsure how to incorporate it. The main goal is to show that point R is located on the circumcircle of OPQ.
  • #1
jdinatale
155
0

Homework Statement


brahmah.jpg



The Attempt at a Solution



So my first thought is that the only way to solve this problem is to apply a characterization of a cyclic quadrilateral. We know that the perpendicular bisectors of a cyclic quadrilateral are concurrent. So here's my thoughts: Construct triangle OPQ. The perpendicular bisector of OP is located on the midpoint of the radius of C(O, OP) and the perpendicular bisector of OQ is located on the midpoint of C(O, OQ). Since OPQ is a triangle, we know these perpendicular bisectors have to be concurrent, and the intersection is the center of the circumcircle. Now we just have to show that R is also located on the circumcircle of OPQ.

The part in the hypothesis about the circles being concentric seems like it is relevant, but I'm not sure how to incorporate it other than what I mentioned earlier.
 
Physics news on Phys.org
  • #2
I'm just not sure how to show that R is also located on the circumcircle of OPQ. Any advice would be appreciated!
 

Related to Proving a quadrilateral is cyclic and finding the radius of the circle

1. How do you prove that a quadrilateral is cyclic?

To prove that a quadrilateral is cyclic, we can use the following theorem: A quadrilateral is cyclic if and only if the sum of opposite angles is 180 degrees. This can be proven using the properties of inscribed angles and central angles of a circle.

2. What is the significance of a quadrilateral being cyclic?

A cyclic quadrilateral has all four vertices lying on the same circle. This means that the quadrilateral has a circumcircle, which is a unique circle passing through all four vertices. This property can be useful in various geometric proofs and constructions.

3. How do you find the radius of the circle containing a cyclic quadrilateral?

The radius of the circle containing a cyclic quadrilateral can be found using the following formula: r = (abc)/(4√(s(s-a)(s-b)(s-c))), where a, b, and c are the side lengths of the quadrilateral and s is the semiperimeter (s = (a+b+c)/2). This formula is derived from the extended law of sines.

4. Can a quadrilateral be cyclic if its opposite angles are not supplementary?

No, a quadrilateral can only be cyclic if its opposite angles are supplementary (add up to 180 degrees). If the opposite angles are not supplementary, the quadrilateral cannot be inscribed in a circle.

5. Is there a relationship between the radius of the circle and the side lengths of a cyclic quadrilateral?

Yes, there is a relationship between the radius of the circle and the side lengths of a cyclic quadrilateral. The radius is inversely proportional to the side lengths, meaning that as the side lengths increase, the radius decreases, and vice versa. This relationship can be seen in the formula for finding the radius of a cyclic quadrilateral (see question 3).

Similar threads

Orthoprojection of circle onto a plane
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Euclidean and non Euclidean geometries problems
    • Precalculus Mathematics Homework Help
Replies
4
Views
3K
Angular velocity of a connecting rod on a rotating wheel.
    • Calculus and Beyond Homework Help
Replies
4
Views
7K
Proving Perpendicular Lines in a Quadrilateral
    • Precalculus Mathematics Homework Help
Replies
3
Views
3K
I Three inertial particles: Separating the twins from the paradox
    • Special and General Relativity
2
Replies
40
Views
2K
Portion of Altitude of a Triangle Inscribed in a Circle
    • Precalculus Mathematics Homework Help
Replies
9
Views
3K
Angles inscribed in circles part 1
    • Precalculus Mathematics Homework Help
Replies
6
Views
2K
Calculating the Speed of Light in a Revolving Lighthouse
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
MHB Find the equation of the polar
    • Linear and Abstract Algebra
Replies
8
Views
2K
MHB Geometry Problem: Find Shaded Area Given Triangle & Inner Circle
    • General Math
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top