Problem 3

Consider all triangles

ABC

with a fixed base

AB

and whose altitude from

C

is a constant

h

. For which of these triangles is the product of its three altitudes a maximum?

Step 1 of 5: Multiply the three area identities

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AB\cdot h\cdot AC\cdot h_b\cdot BC\cdot h_a=8[ABC]^3=(AB\cdot h)^3

Detailed analysis

Let $h_a,h_b$ be the altitudes from $A,B$ . The equal area expressions are $AB\cdot h=AC\cdot h_b=BC\cdot h_a=2[ABC]$ . Multiplying them gives the displayed identity. Since $AB$ and $h$ are fixed, its right side is constant.