An Application of Triangle Geometry

I am always on the lookout for attractive, simple, and real applications of synthetic Euclidean geometry to share with my students. That's why I enjoyed an article in the March 2011 issue of The College Mathematics Journal by Robert K. Smither, "The Symmedian Point: Constructed and Applied". The symmedian point of a triangle is the intersection of the three symmedians, where a symmedian is the reflection of a median of the triangle in the angle bisector (at the vertex through which the median passes). See the picture below, where the dashed lines are the medians, the green lines are the angle bisectors, the red lines are the symmedians, and P is the symmedian point of triangle ABC.

Smither worked for the Navy in the post-WWII era, well before computers were widely available. His job was to design a system for locating mines that might be dropped by aircraft on a harbor. As a mine hit the water, it would be observed from three stations. On a map, rays would be drawn from each station location in the direction in which the mine had been sighted. In theory, the three rays would be concurrent at the location of the mine. In practice, due to measurement error, the three rays would not be concurrent, but would form a triangle ABC. The most likely location of the mine was assumed to be the point for which the sum of the squares of the distances to the sides AB, BC, AC was minimum.

The hand calculations required to locate the point in question using analytic geometry were horrendous and error-prone. By examining the results, Smither was led to rediscover the symmedian point, which turns out to be the point that minimizes the sum of the squares of the distances to the sides of the triangle. He also discovered a neat method of constructing this point, which is easier than using the definition and appears to be original.