Let ABC be an acute triangle with orthocenter H. Let D, E, and F be the feet of the altitudes from A, B, and C respectively.
Observation. Consider the sum formed by adding the two right hand parts of the equations:
This DOES NOT prove our results but it certainly adds something. Does it suggest a possible strategy for proofs?
1. State and prove a corresponding result where P is an interior point to Triangle ABC.
PF/AD + PG/BE + PH/CF = 1
2. Generalize your result to include P as a point exterior to the triangle.
3. Is the result trivial when P is on the triangle?