Consider an acute triangle . Let
denote the
orthocenter and let
,
,
be the feet of
the perpendiculars of
,
, and
respectfully.
Then:
,
and
Proof:
Let ,
,
,
represent the
area of
,
, and
respectfully. So
we have that
,
,
,
Now we also have that
(By
substituting previous values of
,
,
,
)
Now we note that:
(1)
(2)
(3)
By substituting each new altitude representation into our previous result gives us
Which is what we wanted!
Notice if is an obtuse
triangle our relation no longer holds since the orthocenter
lies outside the
triangle
.
Nevertheless, we can now consider the triangle , which has orthocenter
, thereby reducing to the previously proven case!