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 3 of 5: Express the side product using sin C

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(\sin C)AC\cdot BC=BC\cdot h_a=2[ABC]

Detailed analysis

The area formula using the two sides adjacent to $C$ gives $[ABC]=\tfrac12 AC\cdot BC\sin C$ . Equivalently, $(\sin C)AC\cdot BC=2[ABC]=BC\cdot h_a$ , which is constant because the area is fixed. Therefore $AC\cdot BC$ is minimized exactly when $\sin C$ is maximized.