 Hello and how are you all today? My name is Priyanka. The question says if two circles intersect at two points Proof that they're sent to lie on the perpendicular bisector of the common chord. Now this is the figure that will help us in proving the question. We're given that two circles with center O and O dash are intersecting at points P and B. Right, AB is the common chord to these circles and let O O dash Intersect AB at So we need to prove that O O dash is the perpendicular bisector of AB. So let us draw O A OB O dash A and O dash B that I need to draw. Right Let us start with our proof. Now in triangle O A O dash and O B O dash we have O A is equal to OB and O dash A is equal to O dash B because they are the radii of the same circle. Right and further O O dash is equal to O O dash that is common. So therefore triangle O A O dash is congruent to triangle O B O dash by SSS congruency criteria. Right, so I can say that therefore angle A O M is equal to angle B O M and that is also because angle A O O dash is equal to angle A O M angle B O M is equal to angle B O dash. Right further in triangle A O M and B O M these are the two triangles which we are talking about O A is equal to OB this is the radii of the same circle angle A O M is equal to angle B O these were the two triangles which we are talking about. O A is equal to OB that is the radius of the same circle angle A O M is equal to angle B O M that we have proved in first and O M is equal to O M that is common. So therefore triangle A O M is congruent to triangle B O M and therefore and that is by S A S congruency criteria and from that we can say that A M is equal to B M and angle A M O is equal to angle B M O they both by CP CT but as you can see that angle A M O plus angle B M O will be equal to 180 degree that is A M O this angle plus this angle has to be equal to 180 because they are forming a straight angle and we are also given that they are equal to each other. So therefore twice angle A M O is equal to 180 that is angle A M O will be equal to 90 degree. Since angle A M O is equal to angle B M O they both will be equal to 90 degrees since they both are 90 degree each. So what does that prove exactly? It proves that hence O O dash is the perpendicular bisector of A B as A M is equal to M B as proved above and they both are equal to 90 degrees also. So this proves that it is their perpendicular bisector. So this completes the session. Hope you enjoyed it and take care.