Concyclic Points – Proof

Table of Contents

• Concyclic Points Proof
• Summary
• Did You Know?
• What’s Next?

In the previous segment, we learnt the concept of Concyclic points. In this segment, we will understand and prove a theorem based on concyclic points.

Concyclic points proof

The concyclic points theorem states that if a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points are concyclic.

Let us see how we can prove this theorem.

Given four points A, B, C and D, such that C and D lie on the same side of AB.

If angle ACB is equal to angle ADB, we have to prove that exactly one circle passes through these four points.

Figure 1

We use indirect proof to prove that A, B, C and D are concyclic. That is we assume that point D does not lie on the circle passing through points A, B and C.

We know that a circle passes through 3 non-collinear points. So, construct a circle that passes through points A, B and C and meets AD at point D’. Join B and D’ as shown below.

Did You Know?

Concyclic points are also called collinear points. This word comes from the Latin word collineare, meaning “to be in a line.”