Chaitanya's Random Pages

January 9, 2012

Extremal points of a triangle

Filed under: mathematics — ckrao @ 11:41 am

In this post we summarise some of the points of a triangle that maximise or minimise various functions on the plane. This is on a similar theme to an earlier post on optimisation problems given a point inside a given angle. Many of the results are proved in [1].

Firstly we introduce some notation and terminology. Let ABC be a given triangle with side lengths a = BC,b = AC, c = AB and let P be a variable point in the plane. Let x, y, z be the distances from P to the sides AB, AC, BC respectively. The feet of the perpendiculars from P to the sides of ABC form a pedal triangle. If AP, BP, CP meet sides BC, CA, AB in points D, E, F, then \bigtriangleup DEF is known as the Cevian triangle of P.

Functions involving distances to the vertices

  • \displaystyle \min |PA| + |PB| + |PC|: P is the Fermat point of ABC. If the largest angle is 120 degrees or more, the Fermat point is at the vertex of this angle, otherwise we can construct it by drawing an equilateral triangle ABD external to ABC, then finding the intersection of CD and the circumcircle of ABD.
    In this figure one can show that PA + PB + PC = CD.
  • \displaystyle \max \min \{|PA|, |PB|, |PC|\}: if ABC is not obtuse, P is the circumcentre of ABC. Otherwise it is the intersection of the longest side of ABC with the perpendicular bisector of its middle-length side.
  • \displaystyle \min \{aPA + bPB + cPC\}: P is the orthocentre H of ABC, with minimum value given by four times the area of ABC.
  • \displaystyle \min PA^2 + PB^2 + PC^2: P is the centroid G of ABC with minimum value given by (a^2 + b^2 + c^2)/3.
  • \displaystyle \min aPA^2 + bPB^2 + cPC^2: P is the incentre I of ABC with minimum value given by abc.
  • \displaystyle \min xPA^2 + yPB^2 + zPC^2: This generalises the previous two expressions: P is the point described by the position vector (xA + yB + zC)/(x+y+z) with minimum value equal to (a^2yz + b^2xz + c^2 xy)/(x+y+z).
  • \displaystyle \min PA.PB.c+ PB.PC.a + PC.PA.b: P is the orthocentre H of ABC with minimum value given by abc.

Functions involving distances to the sides

  • \displaystyle \min x + y + z: P is the vertex of the largest angle of ABC.
  • \displaystyle \max xyz: P is the centroid G of ABC.
  • \displaystyle \max xy + yz + zx: P is the Mittenpunkt of ABC, given by the position vector [a(b+c-a)A + b(c + a - b)B + c(a + b - c)C ]/[2(ab + bc + ca) - (a^2 + b^2 + c^2) ]. This and the previous fact can be proved using Lagrange multipliers using the constraint that ax + by + cz is constant (equal to twice the area of ABC). The Mittenpunkt is the point of intersection of lines joining the triangle’s excentres (formed by exterior angle bisectors) and midpoints.
  • \displaystyle \max \frac{xyz}{PA.PB.PC}: P is the incentre I of ABC (IMO shortlist 2001)

Functions involving the pedal triangle

  • maximum area of pedal triangle: P is the circumcentre O of ABC.
  • minimum perimeter of pedal triangle: P is the orthocentre O of ABC.
  • minimum pedal radius (radius of circumcircle of pedal triangle): P is the incentre of ABC. See [2] for more details.

Functions involving the Cevian triangle

  • maximum area of Cevian triangle: P is the centroid G of ABC with maximum area given by a fourth of the area of ABC.

 

References

[1] T. Andreescu, O. Mushkarov, L. Stoyanov, Geometric Problems on Maxima and Minima, Birkhäuser, 2006.

[2] V. Naik, Optimization methods in planar geometry, available at http://www.cmi.ac.in/~vipul/olymp_resources/olympiadarticles/geometricoptimization.pdf

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