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suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are




goal: find $t$ for which area of triangle $ABC$ is maximum

first of all,i was thinking that we could find length of each side of triangles,for example


but what about another sides?we can use determinant formula like here

and goal will be find maximum determinant,but could we it?also i have calculate length of $AB$,which is equal $2*cos(t)*cos(3-t)-2*sin(t)*sin(3-t)$ which is i think


or in our case it would be

$2*cos(t-(3-t))=2*cos(2*t-3)$ am on the right way?or could i simplify way of solution? EDITED: so rotation matrix in 2D has form

enter image description here

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The shape and size of the triangle doesn't change when you rotate it. So rotate it so that $B$ comes to lie on $(1,0)$, and $C$ on $(-1,0)$. Where does that place $A$ then? – Daniel Fischer Jul 15 '13 at 15:55
$cos(3),sin(3)$ right? or how rotate? – dato datuashvili Jul 15 '13 at 15:58
@DanielFischer The shape changes. – Lord Soth Jul 15 '13 at 15:58
@LordSoth Huh? If you rotate a figure, you get a congruent figure. – Daniel Fischer Jul 15 '13 at 16:00
@DanielFischer Yea I fell into the same trap. You have $\cos(t)$ and $\cos(3-t)$ one point rotates clockwise, the other rotates counterclockwise, that is why the shape changes. – Lord Soth Jul 15 '13 at 16:01
up vote 1 down vote accepted

Hint: All these points lie on the unit circle. In particular, $B$ and $C$ are antipodal, meaning that the line segment $BC$ passes through the origin. Now, forget about point $A$ as in the problem. Where you should put a point $A'$ such that the area $A'BC$ is maximized, where $BC$ is an antipodal line segment? The solution of this problem is to make $A'BC$ a right triangle (You need to prove this). Now, see if your parametric equations form the same right triangle (up to rotations); and by the way, they will.

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but task is about determine $t$ – dato datuashvili Jul 15 '13 at 16:06
I know; I think this is enough hint to keep you going, and you should easily be able to find $t$. Why don't you spend some time on the problem with this hint and let me know if you still have a problem. – Lord Soth Jul 15 '13 at 16:09
i found length of $BC=2$ ,but now i should use Pythagorean theorem to find length of $A'B$ right and point $A'$ itself yes ?,but it will introduce another variables,maybe much more right? – dato datuashvili Jul 15 '13 at 16:15
OK, forget about the question. I give you two antipodal points on the unit circle, say $B$ and $C$. As you said, $|BC| = 2$. Your task is to find a point $A'$, again on the unit circle, such that the area $A'BC$ is maximized. Where you should put the point $A'$ and what is the resulting area? – Lord Soth Jul 15 '13 at 16:28
antipodal means they are on they same line?on diameter? by 180 degree – dato datuashvili Jul 15 '13 at 16:31

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