Question

Consider two triangles Δ??? and Δ??? that have common side ??, and the circumcentre, orthocentre and...

Consider two triangles Δ??? and Δ??? that have common side ??, and the circumcentre, orthocentre and centroid of both triangles. Draw a line connecting the circumcentres, a line connecting the orthocentres and a line connecting the centroids. Which of these lines are parallel to ??? Answer this by providing appropriate diagrams drawn in GeoGebra (or appropriate geometry program) and accompanying worded explanations as appropriate. You may conveniently choose coordinates for A,B,C,D, so long as Δ??? and Δ??? are not similar nor congruent.

Line connecting cenntroid (here line IJ) is parallel to CD

Explanation:

Below figure we have 2 triangle ABD, and ABC with AB common side

 Line joining intersections Centroid I (line l, m are median of 2 sides) J (h,k are median of 2 sides) IJ Circumcenter L (line p, n are perpendicular bisectors of 2 sides) M (line q, n are perpandicular bisectors of 2 sides) LM Orthocenter N (line t, a are altitudes of 2 sides) P (line b, c are altitudes of 2 sides) PN

We can clearly see that line joining the centroid IJ is parallel to the side CD

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