Happy Indigenous Peoples' Day! You can also feel free to celebrate some Italian guy paid by Spain to attempt and fail to get to India, if that's what you're into. Here's the Problem of the Week, which has to do with neither of them.

Figured out how to show this? Stumped? You can find my work-through of this one after the break.

While you might at first jump to analyzing this as an optimization problem in the style of calculus, you can actually do it wholly geometrically, as I've done it below:

Did you solve the problem the way I did, or another way? Tell us how you did it in the comments!

First method:

ReplyDeleteLet's denote S: area of triangle ABC

O is the center of the unit circle

If we fixed the segment BC and let A move on the circle then the area of ∆ABC gets max when AO ⊥ BC

In the similar manner, BO ⊥ AC and CO ⊥ AB

=> ∆ABC is an equilateral triangle