mixtilinear incircle

2025 March 05

Problem 1. Show that:
  1. $K$, $I$, $L$ are collinear, and $I$ is the midpoint of $KL$;
  2. If $M$ is the arc midpoint of the major arc $BC$, then $T$, $I$, $M$ are collinear;
  3. $B$, $K$, $I$, $T$, are concyclic;
  4. $T,D,A_1,M_A$ are concyclic;
  5. If $N$ is the second intersection of line $AD$ with $(TDA_1M_A)$, then the reflection of $N$ over the angle bisector of $\angle A$, $P$, lies on the circumcircle;
  6. $\triangle BTA\sim \triangle DTC$;
  7. If $T’=AT\cap KL$, $M’$ is the midpoint of $BC$, and $S=M’I\cap AT$, then $S$ is the midpoint of $AT’$.
Problem 2. Given triangle $\triangle ABC$, let the tangency point of the $A$-mixtilinear incircle and the circumcircle be $T$. There exists a circle tangent to the circumcircle at $A$, and the incircle at $P$. Show that $A$, $P$, $T$ are collinear.
Problem 3. Let $R$ be a point on the minor arc $BC$, and let $R_1R_2$ be the polar of $R$ in the incircle. Then let $R_1’=BC\cap RR_1$, $R_2’=BC\cap RR_2$. Show that $(RR_1’R_2’)$ passes through $T$.\footnote{The quickest solution will use 1(b).}
Problem 4. Let the tangency point of the $A$-mixtilinear incircle with the circumcircle be $T_A$, and similarly define $T_B$ and $T_C$. Show that $AT_A$, $BT_B$, and $CT_C$ concur. (This point is called $X_{56}$, the exsimilicenter of the circumcircle and incircle.)