Difference between revisions of "1992 OIM Problems/Problem 3"
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Construct the triangle in the cartesian plane as shown above with the shown vertices coordinates. | Construct the triangle in the cartesian plane as shown above with the shown vertices coordinates. | ||
| − | Point <math>P</math> coordinates is <math>(P_x,P_y)</math> and | + | Point <math>P</math> coordinates is <math>(P_x,P_y)</math> and <math>P_x^2+P_y^2=r^2=\left( \frac{1}{\sqrt{3}} \right)^2=\frac{1}{3}</math> |
| − | <math> | + | Let <math>a, b, c</math> be the distances from the vertices to point <math>P</math> and <math>S</math> the sum of the squares of those distances. |
| + | |||
| + | Part a. | ||
| + | |||
| + | <math>a^2=(P_x-1)^2+\left( P_y-\frac{1}{\sqrt{3})} \right)^2=P_x^2+P_y^2-2P_x+\frac{2}{\sqrt{3}}P_y+\frac{4}{3}</math> | ||
| + | |||
| + | Since <math>P_x^2+P_y^2=\frac{1}{3}</math>, | ||
| + | |||
| + | <math>a^2=-2P_x+\frac{2}{\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 20:42, 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 and 
the sum of the squares of those distances.
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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