Let $I$ be the center of the inscribed circle of the triangle $ABC$. The inscribed circle is tangent to sides $BC$ and $AC$ at points $K_1$ and $K_2$ respectively. Using a ruler and a compass, find the center of excircle for triangle $CK_1K_2$ which is tangent to side $CK_2$, in at most $4$ steps (each step is to draw a circle or a line).
(Hryhorii Filippovskyi, Volodymyr Brayman)
To complete the task in 4 steps, one needs to prove the two claims: $C K_1 = CD $ and $CD$ is parallel to $K_1 K_2$, where point $D$ is the excenter that we are looking for. The proof is simply illustrated in the attached diagram using angluar relation in isoceles triangle and external angular bisector. Therefore we can use compass-ruler to find the point D in 4 steps as follow:
Use 3 steps to draw a line $L$ that passes $C$ and is parallel to $K_1 K_2$ with angle copy method; use compass to draw a circle $P$ centered at $C$ with radius equal to $C K_1$. Then the intersection of line $L$ and circle $P$ is the excenter of $CK_1K_2$ touching $CK_2$. In total there are 4 steps.
