# A generalization of the Sawayama-Thebault theorem

1. Introduction

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 $$I'$$