 Hello and welcome to the session. In this session we discussed the following question which says give a geometrical construction for finding the fourth point lying on a circle passing through three given points without finding the center of the circle justify the construction. Now let's move on to the solution. First we consider let A, B, C be the given points and these points A, B, C lie on the circle with center O. We need to locate a point say point D such that this point D also lies on the circle. So first of all with B as the center and radius equal to AC we draw an arc. So we have drawn this arc taking B as the center and radius equal to the measure of AC. Then in the next step with C as the center and radius equal to AD we draw such that it intersects the previous arc. So this arc is drawn taking CRC center and radius equal to AD. Let this point of intersection of the two arcs remark that the point D then we say that this D is the desired point such that it lies on the circle with center O. Now for the justification of this construction that we have done we will join BD and CD. Next we will consider the triangles ABC and DCB. Now in these two triangles we have decided AB of triangle ABC is equal to decide DC of the triangle DCB. Since as you know that in the construction we have drawn an arc taking CRC center and radius equal to AB. So from this construction we get AB is equal to DC. Next we have that the side AC of triangle ABC is equal to decide DB of triangle DCB. This is again by construction. Since with B as the center and radius equal to AC we have drawn an arc. So this is also by the construction. Then BC is equal to CB that is the common side. Therefore we say that triangle ABC is congruent to the triangle DCB by the SSS congruence rule. And this implies that angle CAB is equal to the angle BDC. This is by the CPCT that is corresponding parts of congruent triangles. Now we know that a line segment joining two points obtains equal angles to other points lying on the same side of the line containing the line segment. The four points lie on a circle. So according to this result that we have just stated we get that the line segment joining the points B and C subtend equal angles angle CAB and angle BDC at the two other points A and D lying on the same side of the line containing the line segment BC. So these four points that is A, B, C, D are consyclic. They lie on a circle. So hence proved this complete C-session. Hope you have understood the solution for this question.