Geometric Optimization
from the Asian Pacific Mathematical Olympiad

The applet below illustrates a Problem 3 from the 2009 Asian Pacific Mathematical Olympiad:

Consider all the triangles ABC which have a fixed base BC and whose altitude from A is a constant h. For which of these triangles is the product of its altitudes a maximum?


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Solution

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Copyright © 1996-2017 Alexander Bogomolny

Solution by
Vo Duc Dien

Let A', B' C' be the feet of the altitudes from A, B, C, respectively. The problem asks for the product AA'×BB'×CC' to be maximum. But AA' is constant, so BB'×CC' must be maximum.

The area of the triangle ABC is also constant since base BC is fixed.

Twice the area of triangle ABC = AA'×BC = BB'×AC = CC'×AB. From there, BB'×AC×CC'×AB, which is the square of twice the area of triangle ABC, is also constant. Regrouping, (BB'×CC') × (AC×AB) is a constant; thus for BB'×CC' to be maximum, AB×AC has to me minimum. The problem reduces to finding a location of A for which AB×AC is minimum.

Let R be the radius of the circumcircle of ΔABC, and a, b and c as the lengths of its sides, as usual. There is a formula

4R Area( ΔABC) = abc.

But, as we know, Area( ΔABC) is fixed, so the product abc of the three sides and the circumradius R attain minimum simultaneously. The minimum is achieved for an isosceles triangle when AB = AC.

The applet illustrates this point (Hint 2).


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Let S be the topmost point of the circumcircle of ΔABC and T the position of A for which TB = TC. S is always above T, or, more accurately, SA' ≥ TA'. The circumcenters of the two triangles BCS and BCT lie on the perpendicular bisector of BC and those of their sides. If O is the circumcenter of ΔABC (and hence of ΔBCS) and Ot that of ΔBCT, then OA' ≥ OtA', implying by the Pythagorean theorem, that, say, BO ≥ BOt, or R ≥ BOt, which tells us that T is the position of A for which the circumradius attains the minimum.

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  • Heron's Problem
  • Optimization in Parallelepiped
  • Matrices and Determinants as Optimization Tools: an Example
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