Difference between revisions of "1992 OIM Problems/Problem 3"
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<math>b^2=2P_x+\frac{2}{\sqrt{3}}P_y+\frac{5}{3}</math> | <math>b^2=2P_x+\frac{2}{\sqrt{3}}P_y+\frac{5}{3}</math> | ||
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| + | <math>c^2=P_x^2+\left( P_y-\frac{2}{\sqrt{3}} \right)^2=P_x^2+P_y^2-\frac{4}{\sqrt{3}}P_y+\frac{4}{3}</math> | ||
| + | |||
| + | <math>c^2=-\frac{4}{\sqrt{3}}P_y+\frac{5}{3}</math> | ||
| + | |||
| + | |||
* 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. | * 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. | ||
Revision as of 20:47, 14 December 2023
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:
![\[\frac{\sqrt{3}}{4}\]](http://latex.artofproblemsolving.com/8/c/e/8ce2da9a94e15e4b68d0b8ca5b86bfff6a90f92c.png)
~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 
Let 
be the distances from the vertices to point .
Part a.
Since 
,
- 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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