1992 OIM Problems/Problem 3
Problem
In an equilateral triangle whose side has length 2, the circle
is inscribed.
a. Show that for every point of
, the sum of the squares of its distances to the vertices
,
and
is 5.
b. Show that for every point in
it is possible to construct a triangle whose sides have the lengths of the segments
,
and
, and that its area is:
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
Construct the triangle in the cartesian plane as shown above with the shown vertices coordinates.
Point coordinates is
and
- Note. I actually competed at this event in Venezuela when I was in High School representing Puerto Rico. I got full points for part a and partial points for part b. I don't remember what I did. I will try to write a solution for this one later.
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