Hello, atreyyu!

A sketch would help . . .

Given is a triangle ABC.

A circle with center $\displaystyle D$ is tangent to $\displaystyle AB, BC, AC$

at points that lie outside the given triangle.

Show that $\displaystyle AD$ passes through the circumcircle of $\displaystyle \Delta BCD$.

This is *one* possible diagram . . .

Code:

\
* * * \
* *\
* *
* *
\ /
* *\ /
* * * o B
* D */ \
/ \
* * \
* * \
* */ \
- - - * * * - o - - - - - o - - -
/C A\
/ \

Circle $\displaystyle D$ is tangent to side $\displaystyle BC$

. . and tangent to sides $\displaystyle AB$ and $\displaystyle AC$ *extended*.

(There are two more locations for circle $\displaystyle D.$)

They could have said: "Circle *D* is *externally* tangent to triangle *ABC*."

. .