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. CYCLIC
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.

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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 𝐴𝐸 ∙ 𝐶𝐸 = 𝐵𝐸 ∙ 𝐷𝐸
• One of the lines is tangent to the circle while the other is a secant (middle
figure). In this case, we have 𝐴𝐵2
= 𝐵𝐶 ∙ 𝐵𝐷
• Both lines are secants of the circle and intersect outside of it (figure on the
right). In this case, we have 𝐶𝐵 ∙ 𝐶𝐴= 𝐶𝐷 ∙ 𝐶𝐸

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 𝑆1 and 𝑆2 with centres 𝑐1 and 𝑐2. Let P be a point such that 𝑃𝐴 = 𝑃𝐵. Then the locus of point P is the radical axis.

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EQUATION

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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).