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Paralogic Triangles

At the points where a line cuts the sides of a triangle , draw three perpendiculars to the sides, one through each point of intersection. The resulting three lines intersect pairwise in three points that form a triangle known as the paralogic triangle of . The paralogic and original triangles are similar triangles, and two triangles are also perspective triangles with the line being the perspectrix.

Amazingly, the circumcircles of and meet orthogonally in two points, with one point of intersection being their similitude center, and the other being their perspector (Johnson 1929, p. 258).

Circumcircle, Orthogonal Circles, Perspector, Perspectrix, Similitude Center, Sondat's Theorem

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References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 258-262, 1929.

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Paralogic Triangles

Cite this as:

Weisstein, Eric W. "Paralogic Triangles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParalogicTriangles.html