$\require{begingroup} \begingroup$ $\def\Ge{G_{\mathrm{e}}}$
A cevian through the Gergonne point $\Ge$ divides the triangle into two, whose corresponding incircles are "kissing" (mutually tangent).
For example, given a triangle $ABC$, the points $D,E$ and $F$ are the tangential points of the incircle and the Gergonne point $\Ge$ is the point of intersection of Gergonne cevians $AD,BE$ and $CF$.
Cevian $CF$ splits $\triangle ABC$ into $\triangle CAF$ and $\triangle CFB$ with corresponding centers and radii of the inscribed circles $I_{c1},\,r_{c1}$ and $I_{c2},\,r_{c2}$.
These two incircles are tangent at the point $T_c=I_{c1}I_{c2}\cap CF$, $I_{c1}I_{c2}\perp CF$.
Many properties of the Gergonne point are described in the literature, for example, in
Deko Dekov. “Computer-Generated Mathematics: The Gergonne Point”. In: Journal of Computer-Generated Euclidean Geometry 1 (2009), p. 14
but it seems that I failed to find this one explicitly stated/used.
Question: Is this a (well-) known property of Gergonne cevian/point? Any reference(s)?
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