For triangle centers, I like to let students construct them on geogebra if the lab is open, then notice and wonder about them…and their properties. Ideally, they would explore and investigate their questions and prove/dispute their claims.
A few questions that arose this morning…
Do the 6 triangles created by medians have equal areas?
I wonder if you dilated the incenter for the inscribed circle, would it become the circumcenter of the circumscribed circle?
Another student stated, not always, unless your angle bisectors became the perpendicular bisectors. (When would that happen?) Without that happening, it’s a dilation and translation.
Comparing the areas of the circles and corresponding triangles, a student asked,
Is the area of the circumscribed circle twice the area of the triangle?
Is the area of the inscribed circle…one half the area of the triangle?
Now to explore the questions…
These were all follow-ups to discussing how these constructions would aid in solving various real-life contextual problems presented at the beginning of the lesson.