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Concyclic Points

Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.

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Student-friendly explanation

For three non-collinear points, a circle can always be drawn. For four points, the fourth point must also lie on that same circle. In Class 9 problems, opposite angles adding to 180 degrees is a common test.

How to write this in exams

  1. 1

    Start with the exact idea

    Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.

  2. 2

    Then show how to use it

    Name the quadrilateral using the four points. Find opposite angle pairs. If one pair adds to 180 degrees, write that the points are concyclic.

  3. 3

    Add one concrete example

    If in quadrilateral ABCD, angle A + angle C = 180 degrees, then A, B, C and D are concyclic.

  4. 4

    Avoid this incomplete answer

    Adding adjacent angles and getting 180 degrees does not prove concyclicity; the test uses opposite angles.

Definition

Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.

Example

If in quadrilateral ABCD, angle A + angle C = 180 degrees, then A, B, C and D are concyclic.

Rule to remember

Concyclic test for four points: if one pair of opposite angles in the quadrilateral is supplementary, the four points lie on one circle.

Memory hook

Four on one circle: opposite angles total 180.

Examples and method

Worked example

In quadrilateral PQRS, angle P = 75 degrees and angle R = 105 degrees. Since angle P + angle R = 180 degrees, points P, Q, R and S are concyclic.

Method to apply

Name the quadrilateral using the four points. Find opposite angle pairs. If one pair adds to 180 degrees, write that the points are concyclic.

Diagram support

A diagram is useful but not always necessary; show four labelled points forming a quadrilateral and a circle through them.

How CBSE asks it

Questions may ask students to prove four points lie on a circle using angle sums or cyclic quadrilateral converse.

Avoid common mistakes

Common confusion

Students may say any four points are concyclic. This is not true; the fourth point must satisfy the circle condition.

Common wrong answer

Adding adjacent angles and getting 180 degrees does not prove concyclicity; the test uses opposite angles.

Exam tip

To prove four points concyclic, try to form a quadrilateral and check one pair of opposite angles.

Quick check

If angle A + angle C = 180 degrees in quadrilateral ABCD, what does it show about A, B, C and D?

It shows that A, B, C and D are concyclic because a quadrilateral with supplementary opposite angles is cyclic.

Answer writing and exam use

1-mark answer

Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary.

2-mark answer

Points are called concyclic if they lie on the same circle. Four points can often be tested for concyclicity by checking whether a pair of opposite angles is supplementary. Concyclic test for four points: if one pair of opposite angles in the quadrilateral is supplementary, the four points lie on one circle. If in quadrilateral ABCD, angle A + angle C = 180 degrees, then A, B, C and D are concyclic.

3-mark answer

For three non-collinear points, a circle can always be drawn. For four points, the fourth point must also lie on that same circle. In Class 9 problems, opposite angles adding to 180 degrees is a common test. Concyclic test for four points: if one pair of opposite angles in the quadrilateral is supplementary, the four points lie on one circle. In quadrilateral PQRS, angle P = 75 degrees and angle R = 105 degrees. Since angle P + angle R = 180 degrees, points P, Q, R and S are concyclic. Questions may ask students to prove four points lie on a circle using angle sums or cyclic quadrilateral converse. Adding adjacent angles and getting 180 degrees does not prove concyclicity; the test uses opposite angles.
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