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Mid-Arc Points

There are two different definitions of the mid-arc points.

MidArcPoints

The mid-arc points M_(AB), M_(AC), and M_(BC) of a triangle DeltaABC as defined by Johnson (1929) are the points on the circumcircle of the triangle which lie half-way along each of the three arcs determined by the vertices. These points arise in the definition of the Fuhrmann circle and Fuhrmann triangle, and lie on the extensions of the perpendicular bisectors of the triangle sides drawn from the circumcenter O.

Kimberling (1988, 1994) and Kimberling and Veldkamp (1987) define a different type of mid-arc points as the perspectors of a triangle in connection with arc midpoints on the incircle (not the circumcircle). These points have triangle center functions

alpha_(177)=[cos(1/2B)+cos(1/2C)]sec(1/2A)
(1)
alpha_(178)=[cos(1/2B)+cos(1/2C)]csc(1/2A)
(2)
alpha_(2089)=[-cos(1/2A)+cos(1/2B)+cos(1/2C)]sec(1/2A).
(3)

See also

Arc, Circumcircle Mid-Arc Triangle, Cyclic Quadrilateral, First Mid-Arc Point, Fuhrmann Center, Fuhrmann Circle, Fuhrmann Triangle, Mid-Arc Triangle, Perspector, Second Mid-Arc Point, Third Mid-Arc Point

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References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 228-229, 1929.Kimberling, C. "Problem 804." Nieuw Arch. Wisk. 6, 170, 1988.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. and Veldkamp, G. R. "Problem 1160 and Solution." Crux Math. 13, 298-299, 1987.

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Mid-Arc Points

Cite this as:

Weisstein, Eric W. "Mid-Arc Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Mid-ArcPoints.html

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