How to show $x_1y_1 + x_2y_2 - (x_3y_3 + x_4y_4) > 0$

Question

I have the following information. All are labels/variables.

$x_1 < x_3 < x_4 < x_2$

$y_1 < y_3 < y_4 < y_2$

$x_1 + x_2 = x_3 + x_4$

$x_1 + a = x_3$

$x_2 - a = x_4$

• a > 0
• all x and y values are postive

Is there any possibility for me to show that $x_1y_1 + x_2y_2 - (x_3y_3 + x_4y_4) > 0$ ?

What I've so far is

$x_1y_1 + x_2y_2 - (x_3y_3 + x_4y_4)$

$x_1y_1 + x_2y_2 - ((x_1 + a)y_3 + (x_2 - a)y_4)$

$x_2(y_2 - y_4) + a(y_4 - y_3) - x_1(y_3 - y_1)$

In the above statement, the first 2 parts are positive. But I'm stuck to show that the difference between the first 2 parts and the third part is positive

Answer 1

The last expression can be written as $$x_2(y_2−y_4)+a(y_4−y_3)+x_1(y_3−y_1)$$ Here all the terms are positive, hence the inequality follows.

Hope it is helpful:)