6. Draw a figure showing the equilateral triangle of Morley's Theorem if two angles of the original triangle measure 90 and 40 degrees.

Solution:

8. Draw a figure showing the equilateral triangle of Morley's Theorem if two angles of the original triangle measure 30 and 80 degrees.

Solution:

10. Prove that, in Fig. 4.30b, the segments DI, FG, and EH are concurrent.

Solution:I is on the perpendicular bisector of FE since IFE is an isosceles triangle. But the perpendicular bisector of FE passes through D, since DFE is an equilateral triangle. Thus, DI is an altitude of triangle DFE. The same is true for FG and EH, all three lines are altitudes of DFE and therefore they meet at the orthocenter of triangle DFE.

12. Investigate the figure formed by the intersections of the four pairs of adjacent four-sectors (or quadrisectors) of the angles of a square, if four-sectors are defined to be the rays partitioning the angle and its interior into four congruent angles and their interiors.

Solution: