Let $1<c<1.3$ be real number. Let us consider the triangle $$D=(c,0)(-c,2(c-1))(c-2,-2(c-1))$$

and consider a point $w=(x,y)$

My question is: Find sufficient conditions in which the point w belongs to the region enclosed by the triangle $D$. I have no idea to start.

  • $\begingroup$ Hint: A point $P$ lies on a line segment $AB$ if and only if there is a unique number $t\in[0,1]$ such that $P=(1-t)\cdot A+t\cdot B$. $\endgroup$ – John Wayland Bales Feb 8 at 18:28
  • $\begingroup$ What do you mean by a point $w$ belonging to the triangle $D$? Do you mean that $w$ lies on the triangle or do you mean that $w$ lies in region enclosed by it? $\endgroup$ – feynhat Feb 8 at 18:35
  • $\begingroup$ @feynhat: See the edited question. $\endgroup$ – Germany Feb 8 at 18:47
  • $\begingroup$ @China Such problems are usually done using this simple trick: let the points of your triangle be $A$, $B$, $C$ and let $D$ be the point which you want to check. Then compute the areas of these triangles: $\Delta ABD$, $\Delta ADC$ and $\Delta DBC$, and check if they add up to area of $\Delta ABC$. $\endgroup$ – feynhat Feb 8 at 18:53

Given $\triangle ABC$, for every point $D$ on side $AB$ between $A$ and $B$ there is one and only one number $t\in(0,1)$ such that

$$ D=(1-t)A+tB\tag{1} $$

For every point $W$ on line segment $DC$ lying inside $\triangle ABC$ there is one and only one number $s\in(0,1)$ such that


Therefore, for every $W$ lying in the interior of $\triangle ABC$ there is one and only one pair of numbers $(t,s)\in(0,1)\times(0,1)$ such that


So the coordinates of $W$ can be found by applying equation (3) to the coordinates of $A,\,B$ and $C$.

enter image description here

ADDENDUM Note that an alternate approach would be to express the conditions on $W$ as three linear inequalities. Since, when $c>1$ the point $W$ must simultaneously be above lines $AC$ and $BC$ and below line $AB$, this fact can be expressed as a system of three simultaneous linear inequalities in variables $x$ and $y$ with coefficients being functions of $c$.


P.S.: Writing as answer, since I don't have enough reputation to comment

To visualize the problem, check how the triangle $D$ varies with parameter $1 < c < 1.3$ in this 3d plot. Here, neglect the $z$-axis value.


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