Geometric/algebraic proof of a quadratic

[raphael3d]

Homework Statement

http://img717.imageshack.us/img717/4029/screenshot20110106at123.png

i don't know how to construct an algebraic proof from this or how to attempt it.

thank you

Homework Equations

The Attempt at a Solution

 

[╔(σ_σ)╝]

For an algebraic proof you can complete the square then solve for x.

 

[raphael3d]

yes.

but i mean with the information from the drawing.

there are two similar triangles, therefore QN/2 = NY/QY. but how can one proceed here?

 

[raphael3d]

or put it in another way, can't find more similar triangles. where is the next one?

 

[Tedjn]

There are many similar triangles, such as OPP' with OQQ' and ONQ with OPM and OPN with OMQ where O is the circle center. But I don't know where to go from there yet.

 

[raphael3d]

assuming that O is the center, which means ON=OQ=OM=QP=1, => NQ=PM=1.
but how can one deduce that the cutting point of P'Q' and PQ is the origin, is there some proof?

 

[Tedjn]

How are you able to conclude that ON = OQ for example? In particular, I doubt that P'Q' hits the origin. This would only be true if (by similarity) PP' = QQ', and there doesn't seem to be any reason why that must happen.

 

[raphael3d]

right. was a wrong assumption.

i guess the vital part is to make constructions with Q'Q,PP' and 2, the only known lengths and to equate the ratios of similar triangles to finally have QY=...,QX=...