October 15th, 2010, 06:14 PM  #1 
Senior Member Joined: Apr 2009 Posts: 106 Thanks: 0  Area Problem
Angle C is a right angle and P is a point in its interior. Through P we draw a line k intersecting the sides of the angle at A and B. If S1 and S2 are the respective areas of Triangle APC and Triangle BPC, prove that (1/S1) + (1/S2) is the same for all lines k through P.

October 15th, 2010, 11:16 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,034 Thanks: 1393 
Let PM and PN be altitudes of triangles APC and BPC respectively. 1/S1 + 1/S2 = (S2 + S1)/(S1·S2) = (AC·BC/2)/((AC·PM/2)(BC·PN/2)) = 2/(PM·PN) is independent of k. 

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