Investigating Some Point & Line Reflections

1) Reflecting a point in three concurrent lines
The dynamic sketch below shows three line segments (or lines) AD, BD and CD concurrent at D as well as a point E not on any of the three line segments (or lines).

Construction: Either click on the 'Sequence 4 Actions' button (which will automatically carry out the four construction steps below) or click on the buttons underneath to carry out the first four constructions:
Step 1: Click on the Show Point E₁' button to reflect E around AD.
Step 2: Click on the Show Point E₂' button to reflect E around BD.
Step 3: Click on the Show Point E₃' button to reflect E around CD.
Step 4: Click on the Show Circle button to construct a circle through three of the four points above.

Conjecture 1
What do you notice about this circle? Formulate a conjecture.
Thoroughly check your observation by dragging any of the red points.

Further Construction: Click on the Show More Reflections button to show more reflections of the points around the three lines. What do you notice?

Investigating Point & Line Reflections

2) Reflecting a line in three concurrent lines
In the dynamic sketch above, click on the 'Link to Lines Reflected' button to navigate to this investigation. Instead of a point, this sketch has a line EF not coincident with any of the three concurrent line segments (or lines).

Construction: Either click on the 'Sequence 4 Actions' button (which will automatically carry out the four construction steps below) or click on the buttons underneath to carry out the first four constructions:
Step 1: Click on the Show Line j₁' button to reflect EF around AD.
Step 2: Click on the Line j₂' button to reflect EF around BD.
Step 3: Click on the Line j₃' button to reflect EF around CD.
Step 4: Click on the Show Circle button to construct a circle tangent to three of the four lines above.

Conjecture 2
What do you notice about this circle? Formulate a conjecture.
Thoroughly check your observation by dragging any of the red points.

Further Construction: Click on the Line j₂'' button to show the reflection of Line j₂' around CD. What do you notice?

Challenge
Can you logically explain why (prove that) your observations in Investigations 1) & 2) are true?

Generalize
Can you further generalize (& prove) your observations in Investigations 1) & 2)?