Extremal points of a triangle

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 www.cmi.ac.in/~vipul/olymp_resources/olympiadarticles/geometricoptimization.pdf

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