This question was inspired by
in-triangle-abc-d-is-a-point-on-ac…,
show-that-am2-pp-a.
Cevians $|AD_a|=d_a$,
$|BD_b|=d_b$, $|CD_c|=d_c$
divide $\triangle ABC$ into three pairs of triangles,
($\triangle ABD_a$, $\triangle AD_aC$),
($\triangle BCD_b$, $\triangle BD_bA$),
and
($\triangle CAD_c$, $\triangle CD_cB$)
in such a way that
incircles for each pair have the same inradius,
$r_a,r_b$ and $r_c$, respectively
(in the image the circle centers are marked
with the corresponding radius).
Is there a known name for such cevians?
Something like "incircle bisectors"?
These three cevians, which lengths are defined as
\begin{align}
d_a&=\sqrt{\rho(\rho-a)}
,\quad
d_b=\sqrt{\rho(\rho-b)}
,\quad
d_c=\sqrt{\rho(\rho-c)}
\tag{1}\label{1}
\end{align}
uniquely define
the sides of the triangle:
\begin{align}
a&=\frac{d_b^2+d_c^2}{\sqrt{d_a^2+d_b^2+d_c^2}}
,\\
b&=\frac{d_c^2+d_a^2}{\sqrt{d_a^2+d_b^2+d_c^2}}
,\\
c&=\frac{d_a^2+d_b^2}{\sqrt{d_a^2+d_b^2+d_c^2}}
\tag{2}\label{2}
.
\end{align}
They provide a nice relations to semiperimeter
$\rho=\tfrac12(a+b+c)$
of $\triangle ABC$:
\begin{align}
\rho^2&=d_a^2+d_b^2+d_c^2
\tag{3}\label{3}
,
\end{align}
area
\begin{align}
S_{\triangle ABC}
&=
\frac{d_a d_b d_c}{\sqrt{d_a^2+d_b^2+d_c^2}}
\tag{4}\label{4}
\end{align}
and inradius $r$ of the triangle $ABC$:
\begin{align}
r&=
\frac{d_a d_b d_c}{d_a^2+d_b^2+d_c^2}
\tag{5}\label{5}
\end{align}
as well as the circumradius
\begin{align}
R&=\frac{(d_a^2+d_b^2)(d_b^2+d_c^2)(d_c^2+d_a^2)}
{4d_a d_b d_c(d_a^2+d_b^2+d_c^2)}
\tag{6}\label{6}
\end{align}
Unfortunately, in general, as the image illustrates,
these cevians are not concurrent.
Next, these "incircle bisectors"
introduce three inradii
\begin{align}
r_a&=\frac{r}{1+\sqrt{1-\frac{a}\rho}}
,\\
r_b&=\frac{r}{1+\sqrt{1-\frac{b}\rho}}
,\\
r_c&=\frac{r}{1+\sqrt{1-\frac{c}\rho}}
\tag{7}\label{7}
,
\end{align}
which also demonstrate plenty of nice relations, for example,
\begin{align}
\left(\frac{r}r_a-1\right)^2
+\left(\frac{r}r_b-1\right)^2
+\left(\frac{r}r_c-1\right)^2
&=1
\tag{8}\label{8}
,\\
\left(\frac{r}r_a-1\right)
\left(\frac{r}r_b-1\right)
\left(\frac{r}r_c-1\right)
&=\frac{r}{\rho}
\tag{9}\label{9}
,
\end{align}
\begin{align}
\rho&=
\frac{r r_a r_b r_c}{(r-r_a)(r-r_b)(r-r_c)}
\tag{10}\label{10}
,\\
d_a&=\rho\left(\frac{r}{r_a}-1\right)
\tag{11}\label{11}
,\\
d_a&=
\frac{r r_b r_c}{(r-r_b)(r-r_c)}
\tag{12}\label{12}
.
\end{align}
Angles at the feet of the "incircle bisectors"
$\delta_a=\angle AD_a C$,
$\delta_b=\angle BD_b A$,
$\delta_c=\angle CD_c B$,
defined as
\begin{align}
\cos\delta_a&=\frac{c-b}a
,\\
\cos\delta_b&=\frac{a-c}b
,\\
\cos\delta_c&=\frac{b-a}c
\tag{13}\label{13}
,
\end{align}
also provide some interesting identities, like
\begin{align}
\cos\delta_a+\cos\delta_b+\cos\delta_c&=
\frac{(a-b)(b-c)(c-a)}{abc}
\\
&=\frac{a}b+\frac{b}c+\frac{c}a-\frac{a}c-\frac{c}b-\frac{b}a
\tag{14}\label{14}
,\\
\cos\delta_a\cos\delta_b\cos\delta_c&=
-(\cos\delta_a+\cos\delta_b+\cos\delta_c)
\\
&=\frac{(a-c)(c-b)(b-a)}{abc}
\tag{15}\label{15}
.
\end{align}
\begin{align}
\cos\delta_a\cos\delta_b+\cos\delta_b\cos\delta_c
+\cos\delta_c\cos\delta_a
&=\frac{a}b+\frac{b}c+\frac{c}a+
\frac{a}c+\frac{c}b+\frac{b}a
-\left(\frac{a^3+b^3+c^3}{abc}\right)-3
\tag{16}\label{16}
\\
&=\frac{2r}R-1
\tag{17}\label{17}
,\\
\sin\delta_a\sin\delta_b\sin\delta_c
&=\frac{2r}R
\tag{18}\label{18}
,\\
S&=\tfrac12\,\rho\, R\sin\delta_a\sin\delta_b\sin\delta_c
\tag{19}\label{19}
.
\end{align}
Two of them define the third one, like the angles of triangle:
\begin{align}
\cos\delta_c&=
-\frac{\cos\delta_a+\cos\delta_b}{1+\cos\delta_a\cos\delta_b}
\tag{20}\label{20}
.
\end{align}
Are there any known references?
Famous
Baker's collection of formulae for the area of a plane triangle
does not mention these cevians
and neither do the [wiki entries on
wiki-Triangle,
wiki-Triangle_inequalities
and
Cevian.
Search on Google Scholar
was also futile (did I missed something trivial?).
Also I can not remember seen any reference
of these parameters used as a triplets,
only a single instance, without any special name,
like in already mentioned
in-triangle-abc-d-is-a-point-on-ac…,
show-that-am2-pp-a.
Summarizing the question:
1) Are there any known references,
where such cevians and identities are discussed/mentioned?
2) Is there a known name/notation for such cevians?
Something like "incircle bisectors"?
Best Answer
Accidentally, I've found this open-access reference:
For such cevians they use a term "the congruent-incircle cevians of a triangle".