Morley's Theorem The steps below will direct you in an illustration of Morley's Th...

Morley's Theorem The steps below will direct you in an illustration of Morley's Theorem, mentioned in Section 1.1.

[1] Construct ΔABC using the Segment Tool.

[2] Select vertices C, A, and B in that order (with shift key depressed) and choose Angle from the MEASURE menu. Then, choosing Calculate from that menu, find the ratio m∠CAB/3, which will be displayed on the screen.

[3] Select the displayed ratio (m∠CAB/3 and choose Mark Angle Measurement. Then from the TRANSFORM menu, select point A and choose Mark Center. Construct ray , selected, and choose Rotate, with Rotate By Marked Angle checked. Click OK and the first angle trisector will be displayed. With that tri-sector still selected, choose Rotate again, and the second angle trisector will appear. Hide ray .

[4] Repeat the procedure of Step 3 at vertices B and C, producing the angle trisec-tors of the other two angles. (Select ray not ray , to be rotated about B for the trisectors of ∠B, and select ray for the trisectors of ∠C.)

[5] Select the points of intersection of the angle trisectors closest to the sides of ΔABC and, with the three points selected, choose Segment from the CONSTRUCT menu. This will produce ΔDEF. For extra effect, display the interior of ΔDEF using Polygon Interior from the CONSTRUCT menu, with D, E, and F selected.

[6] Measure the sides of ΔDEF, displaying these measurements on the screen. (Select D and E, and choose Distance from the MEASURE menu, to display the value DE, and repeat this procedure for pairs E, F and F, D.)

Drag vertex B to various positions to see the effect on these measurements and the shape of ΔDEF.