The Second Lemoine Circle

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Through the symmedian point K of a triangle ABC draw lines parallel to the sides of the orthic triangle. The six points of intersection of those lines with the sides of the triangle lie on a circle (the second Lemoine circle) with center K.

Contributed by: Jay Warendorff (March 2011)
After work by: Paul Yiu
Open content licensed under CC BY-NC-SA


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The triangle formed by the intersection of the altitudes with the sides of a triangle ABC is called the orthic triangle of ABC.

The centroid of a triangle is the intersection of the lines drawn from the vertices to the midpoints of the opposite sides.

Let P be a point inside ABC. The reflections of the three lines AP, BP, and CP in the angle bisectors at A, B, and C meet in a point, called the isogonal conjugate of P.

The symmedian point is the isogonal conjugate of a triangle's centroid.



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