 Hi and welcome to the session, let us discuss the following question. Question says, in figure 6.62, two codes A, B and C, D of the circle intersect each other at point P. Outside the circle, prove that triangle P, E, C is similar to triangle P, D, V and P, A multiplied by P, V is equal to P, C multiplied by P, D. This is the figure 6.62. Now let us start with the solution. Here we can see in the figure angle P, A, C plus angle B, A, C is equal to 180 degree as they form a linear pair. So we can write angle P, A, C plus angle B, A, C is equal to 180 degree. You also know that sum of opposite angles of cyclic quadrilateral is 180 degree. Now clearly we can see C, A, B, D is a cyclic quadrilateral. So sum of angles B, A, C and angle B, D, C must be equal to 180 degree since they are opposite angles of this cyclic quadrilateral. So we can write angle B, A, C plus angle B, D, C is equal to 180 degree. Now clearly we can see in these two expressions, sum of both pairs of angles is equal to 180 degree. We can write angle P, A, C plus angle B, A, C is equal to angle B, A, C plus angle B, D, C. Now B, A, C and B, A, C will cancel each other and we get angle P, A, C is equal to angle P, D, C. Now in the first part we have to prove the similarity of triangle P, A, C and triangle P, D, B. So we can write in triangle P, A, C and triangle P, D, B angle P, A, C is equal to angle P, D, B. As we have already shown above, we have already shown that angle P, A, C is equal to angle B, D, C. Now in these two triangles, this angle is common. So we can write angle A, P, C is equal to angle D, P, B. Now by A, A similarity criteria triangle P, A, C is similar to triangle P, D, B. We know if two angles of one triangle are respectively equal to two angles of another triangle, then two triangles are similar. So here clearly we can see two angles of this triangle are equal to two angles of this triangle. So by A, A criteria triangle P, A, C is similar to triangle P, D, B. Now this completes the first part of the question. Now let us start with the second part. We know ratio of corresponding sides of two similar triangles are equal. So here these two triangles are similar. So we can say P, A upon P, D is equal to P, C upon P, B. Here P, A upon P, D is equal to P, C upon P, B. Now this implies P, A multiplied by P, B is equal to P, C multiplied by P, D. If we multiply both the sides of this expression by P, D, P, B, then we get this expression. Now this completes the second part of the given question. So this completes the both the parts of the given question. Hence proved. This completes the session. Hope you understood the session. Take care and have a nice day.