Processes

=Stage 1:= =The longest line that can be drawn in a circle is the diameter, so, the largest possible isosceles triangle must have the diameter on one of its sides. Then, after drawing the triangles out, we found out that the isosceles triangle's height meet at the highest point of the circle, thus, creating a larger triangle than any other scalene triangles=

=Stage 2:=

= When one angle of the triangle is the same, it means to revolve the triangle around the center point (the top of the circle where the highest point of the triangle is) and it will affect the length of the sides of the triangle. Since the triangles are in the same circle, the sum of the length of the two sides of the triangle should stay the same until the rotation of the triangle crosses the middle line of the circle, thus making the triangle smaller because it less resembles a square. = = =

=Main Question:=

=Firstly, we found out that when we lengthen the equilateral triangle’s base or height, the triangle will lose area because the three sides of the triangle is not to the maximum – one factor (height or base) of the triangle will be less than when it is an equilateral triangle because it is less like a square, which has the largest area of all the quadrilaterals given that the perimeter stays the same, but when it is exactly half a square, the triangle is not the biggest. The reason is that, if you draw a square around the triangle, you will find a little tip of the triangle sticking out, it is this, that causes the difference. So, equilateral triangles are the largest possible triangles that you can draw in a circle. We need to compare the area of the area also. The way is to overlap the triangles first in the circle, then halve the circle, halving the triangle in the process, and compare the parts that is sticking out of the area that both triangles has.=