Cyclologic triangles!

Two triangles $A_1B_1C_1$ and $A_2B_2C_2$ are Cyclologic if the circles $(A_1B_2C_2)$, $(B_1A_2C_2)$, and $(C_1A_2B_2)$ are concurrent in a common point. The point of concurrence is known as the Cyclologic center of $A_1B_1C_1$ with respect to $A_2B_2C_2$. If this is the case, then the circles $(A_2B_1C_1)$, $(B_2A_1C_1)$, $(C_2A_1B_1)$ also will be concurrent. The point of concurrence is known as the Cyclologic center of $A_2B_2C_2$ with respect to $A_1B_1C_1$.

Proposition. Let $ABC$ be a triangle. Consider two points, $P$ and $Q$, in the plane of $ABC$. Let $P_aP_bP_c$ and $Q_aQ_bQ_c$ be the pedal triangles of $P$ and $Q$, respectively.  Let $X$, $Y$ and $Z$ be the orthogonal projections of $P$ onto the sides $Q_aQ_b$, $Q_bQ_c$ and $Q_aQ_c$, respectively. Then, $P_aP_bP_c$ and $XYZ$ are cyclologic triangles. In other words, the circles $(P_bP_cY)$, $(P_aP_cZ)$ and $(P_aP_bX)$ are concurrent at a point, and so are circles $(P_cYZ)$, $(P_bXY)$ and $(P_aXZ)$.

Proof. Supose $G$ is the second intersection of circles $(P_bP_cY)$ and $(P_cZP_a)$, then, 

$$\angle{P_aGP_b}+\angle{P_bXP_a}=\angle{P_cGP_a}-\angle{P_cGP_b}+\angle{P_bXP}+\angle{PXP_a}.$$

Notice that $\angle{P_cGP_a}=\angle{P_cZP_a}$ and $\angle{P_cGP_b}=\angle{P_cYP}+\angle{PYP_b}$. By supplementary angles it is easy to realize that $PP_cQ_cZY$, $PZP_aQ_aX$ and $PYXQ_bP_b$ are cyclic pentagons. So, $\angle{P_cZP}=\angle{P_cYP}$,  $\angle{PYP_b}=\angle{P_bXP}$ and $\angle{P_aZQ_a}=\angle{Q_aXP_a}$. This allows us to re-write the above expression as follows

$$\angle{P_cZP_a}-\angle{P_cYP_b}+\angle{P_bXP_a}=$$

$$\angle{P_cZP}+90^\circ+\angle{P_aZQ_a}-\angle{P_cYP}-\angle{PYP_b}+\angle{P_bXP}+90^\circ-\angle{P_aZQ_a}=180^\circ.$$

Which means that $GP_aXP_b$ is cyclic, so done.

The cyclologic center of $XYZ$ with respect to $P_aP_bP_c$ is clearly $P$.

Remark: the triangle $Q_aQ_bQ_c$ can be arbitrary inscribed in $ABC$. The fact that $Q_aQ_bQ_c$ is a pedal triangle was never used in the proof.

Some more properties of this configuration can be found in the ETC: Cyclologic centers: X(37743) - X(37744).

See also Lozada's cyclologic triangles.