The Sum of Three Ratios in the Triangle

 

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.

Prove:

Prove:

Observation. Consider the sum formed by adding the two right hand parts of the equations:

= 3

This DOES NOT prove our results but it certainly adds something. Does it suggest a possible strategy for proofs?

Hint
More of a Hint. Go here only as a last resort or to confirm that you have better proofs.
EXTENSION. Can you generalize the results for obtuse triangles? Demonstrate.
EXTENSION. Consider a Pedal Point P for an triangle ABC with DEF being the corresponding vertices of the Pedal triangle for P.

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?

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