The following steps in Sketchpad should lead you to discover two very different geometric...

The following steps in Sketchpad should lead you to discover two very different geometric properties at once.

[1] Construct any two congruent segments and such that ∠BAB' is acute. (Rotate and end point B about A through an acute angle of your choice.)

[2] Construct ray making an acute angle with ray and passing through the interior of ∠BAB'.

[3] Using MEASURE, display m∠CBA, and while it is selected, choose Mark Angle Measurement under TRANSFORM.

[4] Construct ray and double-click on B' to make it a center for rotation (or select B' and use Mark Center under TRANSFORM). Rotate ray about B' to ray choosing Rotate By Marked Angle under TRANSFORM. (This will make ∠QB'A ≅ ∠CBA.) Note: It is important that both rays pass through the interior of ∠BAB'. The particular orientation of your figure may require you to go back to Step 3 and select (— 1) • m∠CBA instead. (Use Calculate to multiply by -1.) Then repeat Steps 3 and 4.

[5] Hide ray then select, or construct, the point of intersection D of rays

(a) Drag C and observe the behavior of point D. Use trace under DISPLAY to exhibit the path of point D. What is it? Try to prove your conjecture.

(b) Drag C to such a position that ray lies exterior to ∠BAB'. What happens to the locus of D in this case? Is your knowledge of Euclidean geometry sufficient for you to explain this behavior?