Comparing C,H,G, I of a Triangle and the Medial Triangle

by Paulo Tan

The interesting result of constructing C, H, G, and I of the medial triangle is that the circumcenter of the original triangle lies on the same point as the orthocenter of the medial triangle. Also, as expected, both triangles share the same centroid. In figure 1, C, H, G, and I represents the points for the original triangle and c, h, g, and i represents the points of the medial triangle.

fig1.

Further exploration of the C and h shows that the reason why they lie on the same point is because of properties of parallel and perpendicular lines. The property of circumcenter is that it lies on the intersection of lines that are perpendicular to the midpoint of each side. Thus, the circumcenter in figure 2 is on line j, which is perpendicular to line segment AB and going through the midpoint D.

fig2.

Next, let’s construct at the medial triangle DGF(fig3). By definition line segment AB is parallel to line segment FG. Since line j is perpendicular to line segment AB, then it must also be perpendicular to line segment FG. Thus, the orthocenter of the medial triangle must lie of line j because it goes through the vertex D and line j is perpendicular to line segment FG.

fig3.

Figure 4 shows the construction of the circumcenter of the main triangle, which is on the same point as the orthocenter of the medial triangle.

fig4.

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