The Sawayama-Thebault theorem is one of the best nice theorem in plane geometry. The theorem has a long history. It was published in AMM in 1938 the first solution appeared in 1973 with 24 pages), You can see detail in here and here.
Theorem 1 (Sawayama-Thebault): Through the vertex $A$ of a triangle $ABC$, a straight line $AD$ is drawn, cutting the side $BC$ at $D$. $I$ is the center of the incircle of triangle $ABC$. Let $P$ be the center of the circle which touches $DC$, $DA$ at $E$, $F$, and the circumcircle of ABC, and let Q be the center of a further circle which touches $DB$, $DA$ in $G$, $H$ and the circumcircle of $ABC$. Then $P$, $I$ and $Q$ are collinear
Source of the Figure 1 in here
I have worked for four years to find a real nice generalization of the Thebault theorem and publish in here since two years ago, but has no proof. So I am looking for the proof of the generalization of the Sawayama-Thebault theorem as follows:
2. Generalization of the Sawayama-Thebault theorem
Problem 2: ( See Figure 2). Let $ABC$ be a triangle the red circle through $B$, $C$, the blue circle tangent to $AB$, $AC$ and the red circle. Let $P$ be arbitrary point in the plane, through $P$ construct two tangent lines $PD$, $PE$ to the blue circle. Let $(O_1)$ is the yellow circle tanegnt to the right tangent line $PD$, tangent to the red circle and the sideline $BC$; $(O_2)$ is the yellow circle tanegnt to the left tangent line $PF$, tangent to the red circle and tangent to the sideline $BC$. Then three points $O_1O_2$ through a fixed point when $P$ be moved on a given line.
3. Construct the two yellow circles?
Problem 3: (The generalization of the Sawayama Lemma) The lines $GF$, $HK$ through the incenter of $ABC$ (Figure 2)
4. Construct the point $I'$