NavList:

Finding the symmedian point has received a LOT of attention in Navlist over the years. Geometric constructions exist for finding the symmedian point. One way, for example, is to find the intersection of the triangle's medians after reflection in the corresponding vertex angle bisectors.

While thumbing through the pages of Wikipedia I came across a property of the symmedian that leads to a geometrical construction that is pretty straightforward but I don't remember having seen before. It maybe buried in the Navlist posts of yore but even if so it probably bears repeating.

The property that underpins the construction receives a one sentence reference in MathWorld.

The tangents to the circumcircle of a triangle at two of its vertices meet on the symmedian from the third vertex (Honsberger 1995, pp. 60-61).

Honsberger, R. "The Symmedian Point." Ch. 7 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 53-77, 1995.

As an example take the round of sights

Intercept  Azimuth
1.5 A           10°
3.2 T         122°
5.1 T         255°

Draw the cocked hat as shown by the black triangle.

Find the centre of the circumcircle, shown in green, through the triangle vertices. 
(The intersection of the perpendicular bisector of two of the triangle's sides)

Draw lines tangent to the circumcircle at the triangle vertices as shown in blue and extend them until they intersect
(By standard geometric construction or by plotting square)

Connect the tangent line intersection points with the opposite cocked vertex as shown by the dashed lines. Any pair of these dashed lines intersect at the symmedian (red) point. (Note that in constructing this plot the location of the symmedian point was calculated using the methods described in the Nautical Almanac.)

Robin Stuart

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