An example of basic proportionality would be Pythagoras' theorem, in which he states that for any right angle triangle its hypotenuse when squared is equal to the sum of its squared sides and is given by the formula of:
a2+b2 = c2 whereas a and b are the sides of the right angle triangle with c being its hypotenuse or longest side.
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it can be used to draw tangents from a given point on a circle.
Well, isn't that a happy little question! A non-example of a constant of proportionality would be a relationship where the ratio between two quantities is not always the same. Imagine a situation where the more you paint, the less paint you use each time - that would not have a constant of proportionality. Just like in painting, it's all about finding balance and harmony in the relationships around us.
A theorem is a statement or proposition which is not self-evident but which can be proved starting from basic axioms using a chain of reasoned argument (and previously proved theorems).
A theorem is proven. An example is The "Pythagoras Theorem" that proved that for a right angled triangle a2 + b2 = c2
One example, for those who still use obsolete measurement units, is quarts and gallons.