Given any triangle ABC, let AE, BF, and CG be the altitudes. Notice (figure 6) that triangles AEC and BFC are similar by angle-angle similarity. Angles E and F are both right angles, and the two triangles share angle C. Likewise, triangles AEB and CGB, and BFA and CGA are similar. Therefore, CE/CF = AE/BF, BG/BE = CG/AE, and AF/AG = BF/CG. Ceva's Theorem, says that (AG/BG)(BE/CE)(CF/AF) = 1 if and only if the cevians are concurrent. Then we can substitute the values to get (CG/CG)(AE/AE)(BF/BF) = 1. Therefore, by Ceva's Theorem, the altitudes of any triangle ABC are concurrent.