Can there be a simpler figure than a triangle? And yet we do not always realize where three innocent points in a plane may lead us. The triangle's special lines are already full of surprises. Its three perpendicular bisectors (the lines perpendicular to the sides at their midpoints) are concurrent; so are its three medians (which pass through the midpoint of a side and the opposite vertex) and its three altitudes (the lines through a vertex perpendicular to the opposite side). These define the circumcenter O, the centroid G and the orthocenter H of the triangle. That is all it takes to obtain the most famous collinearity theorem: O, G and H lie on a single line, the Euler line (see page 24). More remarkably still, these three points always occur in that order, and OH = 3 OG.
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Points, circles and quadrilaterals
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The feet of the altitudes, the midpoints of the segments joining the sides to the orthocenter and the midpoints of the triangle's sides all lie on a single circle, known as the Feuerbach circle (see le Cercle, Bibliothèque Tangente 36, 2009). And the center of this circle, whose radius is half that of the circumcircle, lies… on the Euler line, at the midpoint of \[OH\].
Robert Simson was already aware of the Euler line. It was two brilliant French geometers, Jean-Victor Poncelet and Charles-Julien Brianchon, who identified the Feuerbach circle in 1821. But the story did not end there: many others, including Olry Terquem, discovered that further special points lay on this circle.
