The semi-circle has radius 1.
You might start doing something quite common when geometry is taught. Take the drawing above and mark the lengths of nearly all possible intervals:
Have you done this? And now you do not see anything anymore? Fine, so erase all the additions and start the work once more making as many drawings as you wish but avoid putting the names of objects if you can do without. By the way, I need 7 graphical signs to write ``tan(a/2)''. Perhaps it would make sense to denote it in a shorter way: tan(a/2) = w ?
I start examining the drawing above. The measure of the angle below the green interval is a. Why is the measure of the angle on the left equal to a/2?
Why is the blue triangle rectangular one?
Once more I take two rectangular triangles having a/2 as measure of one of their angles. I want to express both x, y as (trigonometrical) functions of a. Obviously, I use Pythagorean theorem. In fact, I use it here three times. When is it used for the third time?
Did you get these results? x2 = 2(1+cos a), y2 = 1+w2 ?
Now, the similarity of triangles of drawings 3 and 4:
gives x/2 = 1/y. Some lines later I will use the fact that yx=2 but just now I use it in this form: x2 = 4/(y2). I can get rid of x2, y2 due to the results obtained before and soon I encounter this formula:
Next, I look for a ``twin formula'' that would express sin a, akin to the one above. I use the similarity of the triangles of drawings 2 and 3:
(sin a)/w = x/y - or putting it in a more usefull way, sin a = (yxw)/(y2). Thus, the twin formula is:
How to ``reverse'' these formulas, expressing w in terms of sin a and cos a ? I return once more to drawings 2 and 3, this time in order to use the similarity of legs of the triangles:
In this manner the Heron angles will be obtained without any effort.