Altitude Ratios of Acute Triangles

By Paulo Tan

This investigation involved proving that in an acute triangle (fig1), where H is the orthocenter,

Fig1 (Click HERE for an interactive GSP file)

Fig2

Now lets see if these ratios hold for obtuse triangles. We know that two of the altitudes of an obtuse triangle will lie outside the triangle (fig3). Thus, the argument we used in the first proof will not hold. Since the second proof uses the conclusion from the first proof, then the second proof will also not hold.

Fig 3 (Click HERE for an interactive GSP file)

After some exploration with GSP, we’ll notice that the sum of

Fig4

RETURN