CYCLIC QUADRILATERALS-converted.pptx

cyclic quadrilaterals that can help your learning more develop. A cyclic quadrilateral is a four sided shape that can be inscribed into a circle. Each vertex of the quadrilateral lies on the circumference of the circle and is connected by four chords. The opposite angles of a cyclic quadrilateral have a total of. 180 ° . What are properties of cyclic quadrilateral?
A cyclic quadrilateral has the following properties: All rectangles are cyclic. A trapezoid is cyclic only if it is isosceles (equal leg lengths, and equal corresponding angle lengths) A cyclic quadrilateral has opposite angles that sum 180 degrees, the total sum of the angles equals 360 degrees. If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle, then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined, they form the vertices of a cyclic quadrilateral.

1. C Y C L I C QUADRILATERALS

2. What is Cyclic Quadrilaterals? A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle. Definition states that a quadrilateral which is circumscribed in a circle is called a cyclic quadrilateral. It means that all the four vertices of quadrilateral lie in the circumference of the circle. In the figure given, the quadrilateral ABCD is cyclic.

3. A B C D E F X Y Z T U V G H J R S L K O M P Q EXAMPLE

4. POWER OF A POINT THEOREM

5. What is Power of a point theorem ? The Power of a Point Theorem is a relationship that holds between the lengths of the line segments formed when two lines intersect a circle and each other. There are three possibilities as displayed in the figures next. The two lines are chords of the circle and intersect inside the circle (figure on the left). In this case, we have One of the lines is tangent to the circle while the other is a secant (middle figure). In this case, we have = Both lines are secants of the circle and intersect outside of it (figure on the right). In this case, we have =  

6. EXAMPLE FIGURES

7. RADICAL AXIS

8. What is Radical Axis ? A radical axis of two circles is the locus of a point that moves in such a way that the tangent lines drawn from it to the two circles are of the same lengths. The radical axis of 2 circles is a line perpendicular to the line joining the centres . Consider two circles and with centres and . Let P be a point such that . Then the locus of point P is the radical axis.  

9. A B C D E F X Y Z T U V G H J R S L K O M P Q EQUATION

10. A B C D E F X Y Z T U V G H J R S L K O M P Q EXAMPLE

11. RADICAL CENTER

12. What is Radical Center? The radical lines of three circles are concurrent in a point known as the radical center (also called the power center). This theorem was originally demonstrated by Monge ( Dörrie 1965, p. 153). It is a special case of the three conics theorem (Evelyn et al. 1974, pp. 13 and 15).