Concyclic Points: Theorem, Proof, Conditions, Properties & Solved Examples

Concyclic or cocyclic points in geometry mean if a group of points sits on the same circle. The distance between each concyclic point and the circle’s center is the same. Concyclic points are 3 points in the plane that do not all fall on a straight line, but concyclic points are not always four or more such points in the plane.

What are Concyclic Points?

Concyclic points can be defined as two or more points that are located on the same circle. Collinear points are another name for concyclic points. This word derives from the Latin verb collineare, which means “to be in a line.”

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The points are non-concyclic if the only way to join them is with an oval or some other shape that is not a circle.

For example: A, B, C, and D are concyclic points in the image below.

What are Concyclic points and Collinear points?

Collinear points and Concyclic points are considered to be similar because Collinear points are another name for concyclic points. This word’s root, collineare, means “to be in a line” in latin.

According to the Concyclic Points Theorem, four points are concyclic if the product of the diagonal and opposite side lengths equals the sum of those two products.

The theorem states that: If the vertices of the cyclic quadrilateral are A, B, C, and D in that order.

Ptolemy’s theorem is used to determine whether four non-linear points are concyclic or not.

The total of the products of the measures of the pairs of opposite sides in a quadrilateral that may be inscribed in a circle is equal to the product of the measures of its diagonals:

Concyclic Points Theorem Proof

Let ABCD be any quadrilateral that is encircled by a circle.

 

In order to prove the proposition,

The inscription angles clearly show that

and

Let E be an AC point so that

then and

……(1)

…..(2)

From equations 1 and 2, we obtain

Hence proved

Concyclic Points Theorem 2

If the segment joining points A and B subtends equal angles at points C and D on the same side of AB, the four points are concyclic.

Take a look at the above image, which demonstrates how C and D are on the same side of AB.

Concyclic Points Theorem 2 Proof:

Draw a circle that goes through points A, B, and C.

Let’s assume it does not go via D. Then, at some point, let’s say E, it will cross AD (or AD extended):

We have the two cases above:

(angles that the same arc AB subtends.)

Therefore, since the circle does cross over D, all four of its points—A, B, C, and D—are concyclic.

Concyclic Points Formula

Ptolemy’s theorem states that four points are concyclic if the product of the diagonal lengths is equal to the sum of the products of the opposite side lengths.

Ptolemy’s theorem can be seen by drawing a quadrilateral that connects the four points, then drawing diagonal lines between opposite corners, as seen in the image below:

So, according to Ptolemy’s theorem, the formula for four points to be concyclic is .

How to Find Concyclic Points?

In geometry, a set of points are said to be concyclic if they lie on a common circle. Two, three, or four points can be identified concyclic using equations. Follow the below points to find whether the given points are concyclic or not:

• If a set of two coordinate points are given, then these two coordinate points will always be concyclic. The easiest way to draw a circle with these two points is to treat the distance between those two points as the diameter of the circle and draw a circle intersecting those two points.
• A set of three coordinate points will almost always be concyclic, unless they are collinear, meaning all lying on the same line. To check if the three points are on the same line, determine the slope between each coordinate. If the slopes are the same, the points are on the same line. The equation to find the slope is as follows: .

A set of four coordinate points will be concyclic, if they follow Ptolemy’s equation. So, according to Ptolemy’s theorem the formula for four points to be concyclic is . Find the distance between each of the points, then insert the distances to determine if the equation is equal. The distance between two points can be determined using the Pythagorean theorem: .

Condition for points to be concyclic

The 4 points are concyclic if the segment between points A and B subtends equal angles at points C and D on the same side of AB.

Concylic Points on a Triangle

The 6 points of intersection of the lines and the triangle’s sides, known as the Lemoine circle, are concyclic if lines are drawn through the Lemoine point and are parallel to the triangle’s sides. the three medians of it.

Concylic Points on a Quadrilateral

A quadrilateral is considered to be cyclic if its vertices are located on a circle. Because the vertices of ABCD (A, B, C, and D) are located on the circle, for instance, ABCD is a cyclic quadrilateral.

Concylic Points on a Polygon

If a polygon’s vertices are all concyclic, it is said to be cyclic.

For example, a regular polygon with any number of sides has concyclic vertices everywhere. A polygon is considered to be tangential if an inscribed circle is tangent to each of its sides; the tangency points are thus concentric on the inscribed circle.

Concylic Points on an Ellipse

In a plane, there are four of these points. In these circumstances, we frequently are unable to create a circle that passes through all four points. But there is a specific scenario when we can draw a circle crossing through all these 4 points.

These particular points are identified as concyclic points since they are situated on the same circle. It’s significant that there is a circumstance where an ellipse can be drawn through these points.

Let’s consider four common concyclic points for a circle and an ellipse A, B, C, and D.

Concyclic Points on a Quadrilateral

A cyclic quadrilateral is a quadrilateral whose all four vertices are concyclic, i.e. all four vertices lie on a circle, is a concyclic quadrilateral. For example, ABCD is a cyclic quadrilateral since the vertices A, B, C and D lie on the circle. Points A, B, C and D are concyclic points.

Concyclic Points on a Polygon

A polygon is defined to be cyclic if its vertices are all concyclic. For example, all the vertices of a regular polygon of any number of sides are concyclic. In the figure given below, the pentagon ABCDE is inscribed in the circle and the circle is circumscribed around the pentagon.

What is the Difference Between Cyclic and Concyclic?

Concyclic is if all the points are in the same circle. Any two points are always concyclic, three or more points depend on the relationship of the points. Cyclic is a circle.

In general, a cyclic quadrilateral means a quadrilateral that is inscribed in a circle. That means there is a circle that passes through all four vertices of the quadrilateral. The vertices are said to be concyclic.

Concyclic Points Properties

Concyclic points properties are as follows:

• An endless number of circles can be drawn around a single point.
• There are an endless number of circles that two points can lie on.
• There are an infinite number of points that can be located on a circle.
• Concyclicity exists for all pairs of points.
• Any pair of points can be used to create a circle’s segment or diameter.
• Any three points that are not in a straight line are concyclic.

Applications of Concyclic Points

1. Geometry Problem Solving:
   Concyclic points help solve complex geometry problems, especially those involving circles, angles, and triangles.
2. Proofs in Euclidean Geometry:
   They are used in proving theorems like the cyclic quadrilateral theorem and angle properties in circles.
3. Coordinate Geometry:
   In coordinate geometry, the concept of concyclic points helps to determine whether given points lie on the same circle using equations.
4. Circle Construction:
   Knowing that points are concyclic allows us to construct a circle through multiple points accurately.
5. Olympiad and Competitive Exams:
   Concyclic point properties are commonly used in advanced mathematics problems in Olympiads and entrance tests.
6. Design and Engineering:
   In fields like CAD (Computer-Aided Design), understanding when points lie on the same circular path is important for designing mechanical parts and curves.

Concyclic Points Examples

Example: 1 In the picture below, determine the values of each pronumeral.

Solution: The cyclic quadrilateral’s opposing angles

x+ 91 =180

x+ 91-91=180 -91

x = 89

Also,

y+85 =180

y+85-85 =180 -85

y = 95

Example: 2 In the following diagram, determine the value of each pronumeral

Solution: The cyclic quadrilateral’s opposing angles

2a+95=180

2a+95-95 =180-95

2a =85

Also,

4p+92=180

4p+92-92 =180-92

4p =88

p=22

Example 3: How to prove points are concyclic?

Solution: Follow the below steps to prove that the given points are concyclic. 

• Finding the product of the lengths of the diagonals of the quadrilateral formed by the points.
• Finding the sum of the products of the measures of the pairs of opposite sides of the quadrilateral formed by the points.
• If these two values are equal, the points are concyclic.

Example 4: How to check if four points are concyclic?

Solution: Four points are concyclic if they meet Ptolemy’s theorem. This theorem states that if the product of the diagonals equals the sum of the product of the opposite sides on the quadrilateral of the four points, then the points are concyclic.

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