Think about a triangle ABC and three different triangles (ABD1, BCD2, and ACD3) that share frequent sides with it, and assume that the perimeters adjoining to any vertex of ABC are equal, as proven. The altitudes of the three outer triangles, passing via D1, D2, and D3 and orthogonal to the perimeters of ABC, meet in some extent.
This may be made intuitive by imagining the determine in three dimensions. Fold every of the outer triangles “up,” out of the web page. Their outer vertices will meet on the apex of a tetrahedron. Now if we think about wanting straight down at that apex and folding the perimeters down once more, every of these vertices will comply with the road of an altitude (from our perspective) on the best way again to its unique place, as a result of every follows an arc that’s orthogonal to the horizontal aircraft and to one of many sides of ABC. The result’s the unique determine.
(Alexander Shen, “Three-Dimensional Options for Two-Dimensional Issues,” Mathematical Intelligencer 19:3 [June 1997], 44-47.)