 Hello and welcome to another problem solving session. In this session we are going to deal with this question which says in triangle ABC the bisector of angle B meets AC at B. So this is the bisector so both angles are X. A line PQ so this line PQ is parallel to AC given okay and the points are PR and Q where they are meeting so P is on AB Q is on BC and R is on BD okay we have to show two parts one is PR times BQ is equal to QR times BP this one and second one is AB into CQ is equal to BC into AP okay so the solution looks very straightforward why because there's a parallel line so gives indication of using of use of Thales theorem or basic proportionality theorem and since there is an angle bisector as well so hence the angle bisector theorem will also be used so what is given given is BD BD bisects angle angle B okay and PQ is parallel to AC okay and we have to prove one what is one one is PR into BQ is equal to QR into BP now guys whenever there are product of sides of a given triangle it gives a very straightforward hint that it is nothing but it has been you know manipulated and a fraction or two ratios are written like that so so basically what I mean is this is nothing but PR by RQ or QR is equal to BP by BP by BQ this is what we have to prove PR by QR is equal to BP by BQ which is directly from you know in triangle in triangle BQP since BR is an angle bisector is bisecting I'm saying bisecting angle B right whether it is BD or BR because since R lies on BD so BR is also an angle bisector in triangle BPQ therefore by internal angle bisector theorem we can say BP upon BQ is equal to PR upon QR which is you know the first one is this is what we have to establish so hence so cross multiplying you can say BP into QR is equal to PR into BQ which is what you have to find out in the first case first one let's try to prove the second one the second one says AB into CQ is equal to BC so AB and BC are involved okay so AB and BC are involved and the line is parallel so hence I can say by basic proportionality theorem or BP we can say what can we say we can say BC upon QC or the corollary by BPT's corollary okay BC upon QC will be equal to AB upon AP isn't it now if you simply cross multiply again you will get BC into AP into AP is equal to QC into AB this is very straightforward both the parts were very very straightforward so hence you get the second result as well hence proved it's just that RHS is written in LHS and LHS is written in RHS in the given condition right so AB into CQ is equal to BC into AP hence proved