Perimeter inscribed parallel-hexagon

Exploration:
1) Start with any ΔABC and an arbitrary point D on AB. From D construct a line parallel to AC, to intersect BC at E.
2) Click on the + buttons on the top right of the sketch below in sequence to construct EF // BA, FG // BC, etc.

Conjecture:
1) What do you notice when you draw the line through I parallel to CB?
2) Click on the last + button below the sketch to show the perimeters of ABC and DEFGHI. What do you notice?
3) Make conjectures regarding your observations in 1) and 2). Then carefully CHECK your conjectures by dragging D, or dragging any of the vertices A, B or C.

.sketch_canvas { border: medium solid lightgray; display: inline-block; } Perimeter of inscribed parallel-hexagon

Explanation (proof):
Can you explain why (prove) your two conjectures above are true?

Explore more:
1) Can you generalize further to the perimeters of polygons inscribed similarly for quadrilaterals, pentagons, etc.? Explore!
2) Explore the relationship between the area of the inscribed hexagon and that of the triangle, as well as the areas of the analogous cases for quadrilaterals, pentagons, etc. Go here for some interactive sketches: areas of some inscribed polygons.

Created by Michael de Villiers, 24 February 2016.