fundamental theorem on isogonal lines

Let $\\triangle ABC$ be a triangle and $A X, B Y, C Z$ three concurrent lines at $P$ .If $AX^{\\prime},BY^{\\prime},CZ^{\\prime}$ are the respective isogonal conjugate lines for $A X, B Y, C Z$ , then $AX^{\\prime},BY^{\\prime},CZ^{\\prime}$ are also concurrent at some point $P^{\\prime}$ .

An applications of this theorem proves the existence of Lemoine point (for it is the intersection point of the symmedians):

This theorem is a direct consequence of Ceva’s theorem (trigonometric version).

单词	FundamentalTheoremOnIsogonalLines
释义	fundamental theorem on isogonal lines Let $\\triangle ABC$ be a triangle and $A X, B Y, C Z$ three concurrent lines at $P$ .If $AX^{\\prime},BY^{\\prime},CZ^{\\prime}$ are the respective isogonal conjugate lines for $A X, B Y, C Z$ , then $AX^{\\prime},BY^{\\prime},CZ^{\\prime}$ are also concurrent at some point $P^{\\prime}$ . An applications of this theorem proves the existence of Lemoine point (for it is the intersection point of the symmedians): This theorem is a direct consequence of Ceva’s theorem (trigonometric version).
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