In this exploration we will look at the triangle ABC and its orthocenter, H, and prove this equation
where D, E, and F are the feet of the perpendiculars from A, B, and C respectfully. See picture below
So we first begin with the equation
(where is indicative of the area of these triangles). By dividing the equation we get
At this point we will convert the area of these triangles to equivalent measures using the segments that are labeled within the triangle. The picture will be divided into the three triangles and of course the whole triangle is ABC.
So our equation becomes
which proves our stated equation.
From the above equation it is easy to prove the following statement
First we will start with these stated equations from the triangle diagram above
AH + HD = AD,
BH + HE = BE,
and CH + HF = CF.
Divide the first equation by AD, divide the second equation by BE, and divide the third equation by CF, we get the following
Adding these three equations gives
and since , then we now have
which proves the equation stated at the beginning.