**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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