Problem 1209

31. a) When four lines intersect, four triangles $A B C, A E F, B F D$ and $C D E$ are formed (Fig. 19). Applying the conclusions from the previous problem to triangle $A B C$ and the three points on its sides: $D \in(B C), E \in(A C)$ and $F \in(A B)$, it immediately follows that the circumcircles of triangles $A E F, B D F, C D E$ intersect at the same point $P$. However, on the sides of triangle $B D F$, there are also points: $A \in(B F), C \in(B D), E \in(D F)$. Therefore, the circumcircles of triangles $C D E, A E F$, and $A B C$ also intersect at one point. This point will be the same point $P$, since the circumcircles $C D E$ and $A E F$ intersect at $P$.

b) Let $O_{1}, O_{2}, O_{3}$ be the centers of the circles passing through points $A, P, E, F ; B, F, P, D ; C, E, P, D$ respectively. We will prove that the quadrilateral $O_{1} O_{2} O_{3} P$ is cyclic. Since the line connecting the centers of two circles is perpendicular to their common chord, $\widehat{O_{2}} + F \widehat{P D} = \pi$. But $F \widehat{P D} = O_{1} \widehat{P} O_{3}$, because in triangles $F P D$ and $O_{1} P O_{3}$, $\widehat{F} = \widehat{O_{1}}$, $\widehat{D} = \widehat{O_{3}}$ (compare the arcs on which these angles subtend).

Therefore, $\widehat{O_{2}} + O_{1} \widehat{P} O_{3} = \pi$, and thus the circle passing through the centers $O_{1}, O_{2}, O_{3}$ will also pass through point $P$. Similarly, it can be proven that the circle passing through points $O_{1}, O_{2}, O_{4}$ (where $O_{4}$ is the center of the circle passing through points $A, B, C$) will also pass through $P$.

c) Let $H_{1}, H_{2}, H_{3}, H_{4}$ be the orthocenters of triangles $A B C, A E F, B F D, C D E$, and let $P_{1}, P_{2}, P_{3}, P_{4}$ be the points symmetric to point $P$ with respect to the lines $B C, C A, A B, D E$ (these points and the orthocenters are not shown in the diagram).

Applying the conclusions from problem 29 to triangle

!

Fig. 20 $A B C$, we get that points $P_{1}, P_{2}, P_{3}, H_{1}$ lie on the same line. Now applying the same conclusions to triangle $A E F$, where $(A E) = (A C), (A F) = (A B), (E F) = (D E)$, we get that points $P_{2}, P_{3}, P_{4}$ and $H_{2}$ lie on the same line. By similar reasoning, we can prove that points $H_{3}$ and $H_{4}$ also lie on the same line.