Altitudes and Orthocenters
Given an acute triangle ABC. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully.
a) We know we can calculate the area of triangle ABC via one of the below congruent equations:
Area of Triangle ABC =. This will be very helpful to us shortly.
b) Begin with our original expression.
Multiply each fraction by BC/BC, AC/AC, and AB/AB respectively. All of which are equal to on, so we maintain the validity of the original expression.
We receive the following:
c) Notice that the products AD*BC, BE*AC, and CF*AB are each equal to twice the area of triangle ABC.
The products are equal, so we can write the fractions over a common denominator K, where K is twice the area of triangle ABC.
d) Multiply by K to receive: K = HD*BC + HE*AC + HF*AB
Therefore, = K/K = 1 as desired. QED.
a) We know the following to be true from part (i):
We can also know the following:
HD = AD - AH
HE = BE - BH
HF = CF - CH
Substitute these into to receive
b) Reduce each fraction to receive
c) After reorganizing it we get as desired. QED.
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