 Hi and welcome to the session, let us discuss the following question, question says, in figure 6.63, D is a point on site VC of triangle ABC, such that VD upon CD is equal to AB upon AC. Prove that AD is the pi sector of angle BAC. This is the figure 6.63, now let us start with the solution, now we are given in triangle ABC V is a point on BC, such that VD upon CD is equal to AB upon AC. Now we have to prove that AD is the bi sector of angle BAC or we can say we have to prove that angle BAD is equal to angle CAD. Now let us do one construction, produce AB to E such that AC is equal to AE and join C. Now let us start with the required proof. Now first of all let us consider triangle AEC, in triangle AEC, AC is equal to AE, we know by construction AC is equal to AE. Now this implies angle AEC is equal to angle AC, we know in triangle equal sides have equal opposite angles, so we can write this implies angle AEC is equal to angle AC. Now we know that we are given BD upon CD is equal to AB upon AC, so we can write BD upon CD is equal to AB upon AC, this is given in the question. Now we know AC is equal to AE, so we can substitute AE for AC. Now clearly we can see BD upon CD is equal to AB upon AE implies AD is parallel to CE, by applying converse of basic proportionality theorem we get AD is parallel to CE. We know converse of basic proportionality theorem states that if a line divides any two sides of a triangle in the same ratio then the line is parallel to third side of the triangle. Here clearly we can see in triangle BCE AD is dividing BE and BC in the same ratio, so by converse of basic proportionality theorem AD is parallel to CE. Now we know AD is parallel to CE and BE is transversal, so angle BAD is equal to angle AEC as their corresponding angles, so we can write AD is parallel to CE and BE is transversal, so we get angle BAD is equal to angle AEC. Now again we know AD is parallel to CE and AC is transversal, so angle CAD is equal to angle ACE as their alternate interior angles, so we can write AD is parallel to CE and AC is transversal, so we get angle CAD is equal to angle ACE. We have already proved that angle AEC is equal to angle ACE, let us name this expression as 1 and we have shown angle BAD is equal to angle AEC, let us name this expression as 2 and here we have shown angle CAD is equal to angle ACE, now let us name this expression as 3. Now we know angle AEC is equal to angle ACE from expression 1, we know these two angles are equal, now if these two angles are equal this implies angle BAD is equal to angle CAD, now from expressions 1, 2 and 3 we get angle BAD is equal to angle CAD, now clearly we can see in the figure angle BAD is equal to angle CAD implies AD by 6 angle BAC, so we can write this implies AD by 6 angle BAC, so this is our required proof, this completes the session, hope you understood the session, take care and have a nice day. Thank you for watching, see you in the next video.