#### Do the Two Conjectures hold for Obtuse Triangles?

Notice that H, the Orthocenter, is no longer
inside the triangle. So, insteead of being able to focus on the
three triangles inside of ABC, we now need to focus on **ABC**, **AHC**, and
**HBC**, which are inside of triangle **AHB**.

Considering the areas of the triangles, we
can say

If we divide everything on both sides by the
area of triangle ABC, we obtain 1 on the left side.

Using our sketch we can obtain the following
formulas the areas of our regions.

We can use these formulas to substitute, obtaining

We can simplify this equation to obtain

Unfortunately, this does not match
our desired equation which is
In a way, this makes since, because the orthocenter
was taken outside of the triangle ABC, so the altitudes are longer.
It is amazing that so much of the relationship was retained.

Since the second conjecture builds on the first,
it will fail in the case of an obtuse triangle, as well.

Do the Two Conjectures hold for Obtuse Triangles?

No.

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