SOLUTION: Two circles meet at points P and Q, and diameters P A and P B are drawn. Show that the line AB goes through the point Q. (Probably it is easier to think of drawing the lines AQ a

Algebra ->  Geometry-proofs -> SOLUTION: Two circles meet at points P and Q, and diameters P A and P B are drawn. Show that the line AB goes through the point Q. (Probably it is easier to think of drawing the lines AQ a      Log On


   

Question 565708: Two circles meet at points P and Q, and diameters P A and P B
are drawn. Show that the line AB goes through the point Q. (Probably it is easier to
think of drawing the lines AQ and QB and then showing that they are actually the
same line.)
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!


The easiest solution is probably to draw the segment connecting the centers of the circles (denote Y,Z), as well as segment PQ:


Since PY = (1/2)PA and PZ = (1/2)PB, triangles YPZ and APB are similar with a 1:2 ratio. Additionally, PR = (1/2)PQ (this can be proven by symmetry). Since R lies on YZ, Q must lie on AB.

Or, another way you can prove it is show that the pairs of triangles PRY/PQA and PRZ/PQB are similar. Then, you may let angle PRY = m, angle PQA = m, it follows that angle PRZ = angle PQB = 180-m. Hence, angles PQA + PQB = 180, so A,Q,B are collinear.