Altitudes and an OrthocenterÉand ratios which sum to oneÉor two!

 Gayle Gilbert & Greg Schmidt


Consider an acute triangle .  Let  denote the orthocenter and let , ,  be the feet of the perpendiculars of , , and  respectfully.












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:










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!