It is quite an interesting question to solve. The proof explained in this video tutorial used following concepts. You are recommended to understand these tutorials in case you are not aware of any of these. This question may look difficult but it us not so.

Theorem 10.11 – The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.

Exterior angle theorem – If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.

Quadrilateral – Angle sum property – The sum of all four internal angles of a quadrilateral is always 360 degrees.

If we prove that DBC is a straight line that would mean DC is a line and B lies on it. To prove that DBC is a straight line, we need to prove that angle DBA and angle CBA is 90 degree each. 

Theorem – Angle in a semicircle is a right angle.

There are multiple concepts that one should be aware of before solving this question. The question in itself is quite an easy to solve but would need a good knowledge of some very important concepts of circles and triangles. Watch these video tutorials before solving this question.

Theorem 10.9 – Angles in the same segment of a circle are equal.

Theorem 6.7 – Angle sum property of triangle – The sum of the angles of a triangle is 180°

Theorem 10.11 – The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.

Theorem 10.8 – The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Theorem: Angle in a semicircle is a right angle.

Before solving this question, you must be aware of the following theorem. Watch this tutorial first.

Theorem 10.7 – Chords equidistant from the centre of a circle are equal in length.

To complete a circle we need to know its:

• Center
• Radius

Once we have this information, we can draw the circle. If we are given a part of circle (an arc) then using Perpendicular Bisector Theorem, we can find the radius and centre of the circle and can complete it.

This is relatively bit tricky theorem to prove (not difficult though) in this chapter. There are two prerequisite theorems that you must know before you solve this theorem:

Theorem 10.5 – There is (always) one and only one circle passing through three given non-collinear points.

Theorem 10.11 – The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.

Any four points in a plane are concyclic if these lie on a circle. The quadrilateral formed by joining all these points is called a cyclic quadrilateral. The prerequisite to prove the above mentioned theorem is theorem 10.5 which states that there always exists one (and only one) circle passing through three non-collinear points. It is highly recommended that you should watch the prerequisite theorem first in case you are not aware of this.

Here is the direct link to YouTube video tutorial of theorem 10.5.

If there exists a circle such that all four vertices of a quadrilateral passes through it, then it is known as cyclic quadrilateral. To prove this theorem the prerequisite is the proof of theorem 10.8 which states: “The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.”