Let
be a triangle with
. Its incircle with centre
touches the sides
,
, and
in the points
,
, and
respectively. The angle bisector
intersects the lines
and
in the points
and
respectively. Let
be the foot of the altitude through
with respect to
.
Prove that
is the incentre of the triangle
.
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Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$, and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$.
Prove that $D$ is the incentre of the triangle $XYZ$.
Školjka