Euclidean Geometry

[Guest]

Show that the product of PP1 and PP2 can be expressed as PO^2-r^2 where r is the radius of the circle, P is a pt outside the circle, P1 and P2 lie on the circle. P1, P2, and P lie on the same line that intersect the circle. O is the center

I have the answer of (PP1)(PP2)=(PO-OP1)(PO+OP2)=PO^2-r^2

But i dont know how to prove this. Any help?

 

[pka]

Draw a tangent from P to the circle at B so that O and B are on the same side of line PP<SUB>1</SUB>.
There is a theorem that states: \(\displaystyle PP_1 B = PBP_2 \quad \Rightarrow \quad PB^2 = \left( {PP_1 } \right)\left( {PP_2 } \right)\).

Because the triangle \(\displaystyle \Delta POB\) is a right then \(\displaystyle OP^2 = OB^2 + BP^2\).

Can you finish this?

 

[Guest]

what do the colon signify, and what theorem are u referring to?

 

[stapel]

what do the colon signify

Proportionality.

what theorem are u referring to?

Different books list the theorems in different orders and using different names. You need to look back through your book to find whatever your book calls it.

Eliz.