1. ## [SOLVED] Harmonic ratios

Suppose we have two orthogonal circles and any diameter AC of one circle (say) cuts the other circle at B and D.
The circles are intersecting and the prolonged diameter of the first circle (centre O) also passes through the second centre (O'). Hint: Consider OB.OD...

Prove that B and D divide AC harmonically...
Help?

2. Hi,
can you please restate the problem more clearly?
I don't understand

• why the first sentence ends with "and."
• what is a "diagonal of a circle"
• where's point D, maybe some drawing would be helpful to know what points are in play
• what is actually the question? (:

3. [QUOTE=Taluivren;364954]Hi,
can you please restate the problem more clearly?
I don't understand

Sorry, I realize I wrote it in a hurry...should be better now.

4. It is still very confusing, my guess of what's going on is on the attached picture.

Hint tells us to use power of a point O to the circle with center O', OE^2=OB.OD

AB = OB+OA = OB+OE = OE^2/OD +OE = (OE/OD)(OE+OD)
CB = OC-OB = OE-OB = OE-OE^2/OD

AB/CB = (OE+OD)/(OD-OE)

CD = OD-OC = OD-OE

5. You got it almost right, but the diameters of the circles are colinear, and you need to switch points B and C.

6. Originally Posted by HelenaStage
You got it almost right, but the diameters of the circles are colinear, and you need to switch points B and C.
if you interchange points B and C on the picture, the calculation above will stay correct as it is, showing that B and D divide AC harmonically. Sorry for interchanging them on the pic.

I've drawn the points A,O,O' not colinear on purpose, because for the calculation above we don't need their colinearity at all.
Is everything clear or should the picture be redrawn? (:

7. Originally Posted by Taluivren
if you interchange points B and C on the picture, the calculation above will stay correct as it is, showing that B and D divide AC harmonically. Sorry for interchanging them on the pic.

I've drawn the points A,O,O' not colinear on purpose, because for the calculation above we don't need their colinearity at all.
Is everything clear or should the picture be redrawn? (:
Oh no, I get it now!

Thanks a lot...XD