Euclidean geometry proof concerning circles

[Maluka8]

i really need help with this proof.

suppose two circles intersect at points P and Q. Prove that the line containing the centers of the circles is perpendicular to line segment PQ

 

[ammontgo]

Ok. Draw these two circles on graph paper and prove that there is a 90 degree angle between PQ and the line containing the centers.

 

[HallsofIvy]

Draw radii from the centers of the two circles to the points where the two circles intersect as well as the line connecting the two centers. You should immediately see two congruent triangles- "side-side-side". That tells you that the angles made by the two radii with the line between centers are congruent.

Now draw the line connecting the two intersection points. In either circle now, you have two triangles made by the two radii, the line connecting the centers and the two parts of the line connecting the intersection points. And now you have "side angle side" with the first "side" being the two radii- which are congruent because they are both radii of the same circle- the "angle" is the angle between radii and the line connecting the centers which were just proven congruent, and the final "side" is the line connecting the centers. Since those two triangles are congruent, corresponding parts, in particular the angles where the line connecting the centers intersects the line connecting the intersection points, are congruent. Since those two congruent angles make a straigh line (the line connecting the intersection points), they are right angles.