(1) Construct a regular hexagon 55mm across the flat. Within it inscribe the largest possible isoscelence triangle having it base 40mm long across the flat. (2) Construct a regular octagon within the following direct 45mm across the flat, 50mm across the cornes. (3) construct a regular pentagon given a diagonal AB 60mm long.

(1) To construct the largest possible isosceles triangle within a regular hexagon, you can follow these steps:

Step 1: Draw a regular hexagon with a side length of 55mm. This can be done by drawing six equal line segments, each measuring 55mm, and connecting the endpoints to form a closed shape with six sides.

Step 2: Draw a line segment perpendicular to one of the sides of the hexagon, starting from the midpoint of that side. The length of this line segment should be 20mm, half of the desired base length of 40mm.

Step 3: From the endpoint of the first line segment, draw two congruent line segments, each measuring 40mm, forming the base of the isosceles triangle. These two line segments should be angled towards the adjacent sides of the hexagon.

Step 4: Connect the endpoints of the two base line segments to the vertices opposite to the respective adjacent sides. These lines will complete the isosceles triangle within the hexagon.

By following these steps, you will have constructed the largest possible isosceles triangle within the given regular hexagon.

(2) To construct a regular octagon with specific measurements, you can follow these steps:

Step 1: Draw a line segment measuring 45mm across the flat. This will serve as the base of the octagon.

Step 2: At each endpoint of the base line segment, draw two congruent line segments, each measuring 50mm, forming two right angles. These two line segments should be angled towards each other, forming a V-shape.

Step 3: Connect the endpoints of the line segments you just drew to form a closed shape with eight sides, known as an octagon.

By following these steps, you will have constructed a regular octagon with a base of 45mm across the flat and 50mm across the corners.

(3) To construct a regular pentagon given a diagonal AB measuring 60mm, you can follow these steps:

Step 1: Draw a line segment measuring 60mm, which will serve as the diagonal AB.

Step 2: From one endpoint of the diagonal AB, draw a perpendicular line segment measuring 60mm. This line segment will be one of the sides of the pentagon.

Step 3: Extend the other endpoint of the diagonal AB and mark a point that is 60mm away from the diagonal, forming a right angle with the extended line.

Step 4: Connect this new point to the endpoint of the perpendicular line segment you drew earlier to form another side of the pentagon.

Step 5: Repeat steps 3 and 4 two more times, connecting the new points to form the remaining sides of the pentagon.

By following these steps, you will have constructed a regular pentagon with a diagonal AB measuring 60mm.