1)Given a triangle. From a vertex draw a line to opposite side such that the two smaller triangles have same perimeter. Do the same with the second and third vertex. Show that the three lines intersect in a single point called "Perimecenter"

Here is Diagram from Quick:

2)Given a triangle. From the vertex draw a line to opposite side such that the areas of the two smaller triangles are equal. Do the same with second and third. Show that the three lines intersect in a single point.

3)A physicist trys to show off to a mathemation. He says I found an error in mathematics. Thus, the mathemation says, "What?".

And he does the following:

Consider the set of rationals $\displaystyle \mathbb{Q}$.

Assume that they arecountablethen we can write all of them in a row (for example),

.123123123123123...

.454545454545454...

.121212121212121...

.666666666666666...

................................

Define a new number whose in the following way.

If the bold number is "1" then it is "5" if it is not "1" then it is "4".

Then we obtain,

.5454..................................

This new number is not the first one (because it has a different 1st digit).

This new number is not the second one (because it has a different 2nd digits).

And thus on.

Thus, this number is not contained in the list.

Thus, $\displaystyle \mathbb{Q}$ is uncountable.

The physicists is all smiling thinking he is right. But the mathemation is looking at him like an idiot, why?