Altitudes and Orthocenters

by

Hyeshin Choi

1. I construct triangle ABC and find orthocenter, and label it H. Then

and

Click here to see orthocenter GSP.

Now I want to begin by dividing this triangle into three small triangles. TriangleABC=triangle AHB+triangle AHC+ triangle BHC. Since HF, HE, and HD are the altitudes of each triangle, we can show these statements are true by finding the area of these triangles.

Let M, M1, M2, and M3 represent area of eachtriangle respectively.

M=M1+M2+M3, where

M1=(1/2)(AB)(HF),

M2=(1/2)(AC)(HE), and

M3=(1/2)(BC)(HD)

Then M=M1+M2+M3. Now divide this equation by M to both sides, then we have

1=(M1+M2+M3)/M.

1=M1/M+M2/M+M3/M. Then plug back the M1, M2, and M3. Now we got the proof of original statement.

.

Proof of second statement.

Now we know that HD=AD-AH, HC=CF-CH, and HE=BE-BH.

(AD-AH)/AD+(CF-CH)/CF+(BE-BH)/BE

=1-HD/AD+1-HE/BE+1-HF/CF

=3-[]=3-1=2.