Does it exist?

Take an isosceles triangle with base 200cm and height $\frac {1}{200} cm^2$ . The total length of the rest two sides(equal) must be longer than 200cm so the each side is longer than 100cm. The area of that triangle is $0.5cm^2$ . Therefore the statement is TRUE.

Haha. It really exists! :-D

The correct way to prove this, is to take an isosceles triangle. Fix the base as 101 and the opp side(just a notation) as also 101. Extrapolate the base to form a right angled triangle with the vertex of the opp side and the hypotenuse. From the right angled triangle, we can estimate that critical angle = Inv sin (h/101) . The obtuse angle of the triangle is 180 - critical angle. For the area to be 1 cm^2, the height has to be (2/101) fixing the base as 101. Sub this in the critical angle formula; you will get that as 0.01 degrees approx. So, the obtuse angle of the triangle has to be greater than (180 - 0.01) and lesser than 180. The epic statement :P LHS = RHS and hence proved.

100,100,200 have a area of 0

I could be wrong :) Let be a triangle whose sides are respectively $102 cm$ , $102 cm$ and $103 cm$ . Then, calculate the triangle's area by using the second theorem of trigonometry: $A=a \times b \sin \theta$ . Since the area must be less than $1 cm^{2}$ , we have to solve $\frac{102 \times 102 \sin \theta}{2}<1$ $\boxed{and}$ $\frac{102 \times 102 \sin \theta}{2}<1$ $\boxed{and}$ $\frac{102 \times 103 \sin \theta}{2}<1$ . The first and the second inequalities are equivalent, so just one of them can be solved. The overall solution is $\sin \theta <\frac{1}{5253}$ , that is $0<\theta<\sin^{-1}(\frac{1}{5253})$ $\boxed{or}$ $\pi-\sin^{-1}(\frac{1}{5253})<\theta<\pi$ . Hence, it exists at least one triangle that turns the statement TRUE.

Just by a counter example, choose an isosceles triangle whose base measures 300 cm and height of 1/300 cm

let two sides be 100.0000004cm and the other side 200cm.
So the perpendicular from vertex
=sqrt(100.0000004^2 - 100^2) = 0.008944.
The area = (1/2)200* 0.008944 < 1sq.cm.......................................................................................

There was no need to give the diagram. It was misleading.

I am so good with geometry* that I did this:

"0.5 * 200 * 100 * tan(0.00001) < 1"

and it evaluated to True.

* If you are slow, that was sarcasm.

Area = 1/2 base ht ... Base=100.1 cm, Ht=2 divided by 100.1 = 20/1001 of a cm. Any line meeting the apex will be greater than 100.1, therefore validating the statement.