My original solution was written in Dutch (because I'm from Belgium) so maybe I didn't properly translated it to English. There were some parts that I didn't know how to say mathematically in English, so that could also be a reason why my explanation wasn't very clear. I'll ask my teacher when I see him again, but thank you for your effort!
I wasn't saying that the solution isn't explained mathematically, but that you didn't directly tell us what you were assigned to do. As a result, we don't know what your teacher would consider to be a correct answer. In addition, by mixing together the reason for each step with the process, it took me considerable effort to see what you actually did. This is a matter of style, not just of language. The result is that even if you were right, it was not clear; and what you wrote obscured the place where you didn't do what your teacher probably required, namely to make an exact construction.
Here is my understanding of what you did, stated in such a way that it is clear how you constructed the triangle:
I constructed a circle with radius 4 and extended a radius, M
1M
2, the same distance beyond the circumference to A. Then I constructed tangents to the circle, AP
1 and AP
2, so that angle P1AP2 is 60°, since AM
1P
1 and AM
1P
2 are 30-60-90 triangles. (I presume you have learned how to do this, so you don't need to state the details.)
Then I extended line AP
1, and intersected it with a circle centered at M1 so that BM
1 = 4/sin(15°) = 15.5, and M
1BP
1 = 15°. [This is probably what your teacher objects to, because a proper construction does not use calculated measurements; the correct value for BM
1 is 15.454813..., not 15.5, so you can't construct it exactly without further work.]
Then I constructed a chord P
1P
3 perpendicular to BM
1, so that BP
3 is tangent to the circle.
Finally, I extended tangents BP
3 and AP
2 to meet at C. Triangle ABC meets the requirements.
One way to make this a proper construction is to first construct tangent AP
2 and radius M
1P
3 parallel to AP
2. Then make tangent line BC perpendicular to M
1P
3. This ensures that BCA is a right angle and the circle is tangent to the sides.
You would want to state the proof that each step does what you want, after showing the construction.