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X(5), X(20), cusps of the Steiner deltoid of ABC vertices of the cevian triangle of X(69) infinite points of the Napoleon cubic |
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The cubics passing through the cusps of the Steiner deltoid of ABC, the vertices of the cevian triangle of X(69), the centers X(5), X(20) form a pencil which contains one circum-cubic namely the Steiner-McCay cubic K071. Each cubic of the pencil meets the Euler line at a third point X and then also contains the infinite points of the isogonal pK with pivot the reflection of X about X(5). When X = X(5), the cubic is tangent at X(5) to the Euler line. It is K652 and the pK is the Napoleon cubic. The cubic and the pK meet at six other finite points which lie on a same rectangular hyperbola and all these hyperbolas belong to a same pencil. The tangents to K652 at the cusps of the deltoid concur at X(5) and make 60° angles with one another. In other words, they are the axes of the deltoid. |
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