Assignment 8:  The Orthocenter


Jonathan Sabo

Given triangle ABC. Construct the Orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, and C respectfully. Prove:

First  we will construct the orthocenter of Triangle ABC.  In order to do this we need to find all of the Altitudes of the triangle.  The point where all of these Altiitudes meet will be the orthocenter of the tirangle.  We will call the Orthocenter point H.  
Now we will construct each of the points where the altitude intersects the opposite side of the triangle.  This will be the feet of the perpendiculars.  Points D, E, and F will be the feet of the perpendiculars from A,B, and C respectively.  

First we are going to prove the following.


Now we will prove the following.  

These conditions will only work if the triangle is an acute triangle.  Observe this GSP construction to see what happens if the triangle is obtuse.


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