Problem 1729

1. Define the given elements and their relationships:
- Let $K_1$, $K_2$, and $K_3$ be three pairwise externally tangent circles.
- Let $P_1$ be the tangency point of $K_2$ and $K_3$.
- Let $P_2$ be the tangency point of $K_1$ and $K_3$.
- Let $AB$ be a diameter of $K_3$, where $A$ and $B$ are points on $K_3$ different from the tangency points.
- Let $AP_2$ intersect $K_1$ at point $X$ (for the second time).
- Let $BP_1$ intersect $K_2$ at point $Y$ (for the second time).
- Let $Z$ be the intersection point of lines $AP_1$ and $BP_2$.

2. **Prove that points $X$, $Y$, and $Z$ are collinear:**
- Denote $AP_2 \cap K_1 = X_1$, $AP_1 \cap K_2 = X_2$, $BP_1 \cap K_2 = Y_1$, $BP_2 \cap K_1 = Y_2$, $AP_1 \cap BP_2 = Z_1$, and $AP_2 \cap BP_1 = Z_2$.
- We need to prove that $X_1, Y_1, Z_1$ are collinear and $X_2, Y_2, Z_2$ are collinear.

3. **Prove collinearity of $X_1, P_3, Y_1$ and $X_2, P_3, Y_2$:**
- Let $K_1 \cap K_2 = P_3$.
- We need to show that $X_1, P_3, Y_1$ and $X_2, P_3, Y_2$ are collinear.
- Consider the angles formed by these points:
$$\angle Y_2 P_3 X_2 = \angle Y_2 P_3 P_2 + \angle P_2 P_3 P_1 + \angle P_1 P_3 X_2$$
$$= (180^\circ - \angle P_3 Y_2 P_2 - \angle P_3 P_2 Y_2) + \angle P_2 P_3 P_1 + (180^\circ - \angle P_3 P_1 X_2 - \angle P_3 X_2 P_1)$$
$$= 180^\circ - \angle P_3 Y_2 P_2 - (180^\circ - \angle P_3 P_2 Z_1) + \angle P_2 P_3 P_1 + 180^\circ - (180^\circ - \angle P_3 P_1 Z_1) - \angle P_3 X_2 P_1$$
$$= \angle P_3 P_2 Z_1 + \angle P_3 P_1 Z_1 + \angle P_2 P_3 P_1 - \angle P_3 Y_2 P_2 - \angle P_3 X_2 P_1$$
$$= \angle P_3 P_2 Z_1 + \angle P_3 P_1 Z_1 + \angle P_2 P_3 P_1 - (180^\circ - \angle P_2 Z_1 P_3)$$
$$= \angle P_3 P_2 Z_1 + \angle P_3 P_1 Z_1 + \angle P_2 P_3 P_1 + \angle P_2 Z_1 P_3 - 180^\circ$$
$$= 180^\circ \implies X_2, P_3, Y_2 \text{ are collinear.}$$

4. Prove collinearity using diameters:
- Since $Y_1 X_2$ and $X_1 Y_2$ are diameters of $K_2$ and $K_1$ respectively, we have:
$$\angle Y_1 P_3 X_2 = \angle X_1 P_3 Y_2 = 90^\circ$$
- Therefore, $X_2, P_3, Y_2$ and $X_1, P_3, Y_1$ are collinear.

5. **Prove collinearity of $P_3, Y_1, Z_1$ and $X_2, Z_2, P_3$:**
- Use the fact that $P_1 P_2 P_3 Z_1 Z_2$ is cyclic.
- Show that $P_3, Y_1, Z_1$ are collinear:
$$P_3, Y_1, Z_1 \text{ are collinear} \Leftrightarrow \angle Y_1 P_3 P_1 = \angle Z_1 P_3 P_1$$
$$\angle Z_1 P_3 P_1 = \angle Z_1 P_2 P_1 = \angle B P_2 P_1 = \angle B A P_1 = \angle P_1 X_2 Y_1$$
$$\angle P_1 X_2 Y_1 = \angle P_1 P_3 Y_1 = \angle P_1 P_3 Z_1 \implies P_3, Y_1, Z_1 \text{ are collinear}$$
$$\implies X_1, Y_1, Z_1 \text{ are collinear.}$$
- Show that $X_2, Z_2, P_3$ are collinear:
$$\angle Z_2 P_3 Z_1 = \angle Z_2 P_3 Y_1 = 90^\circ$$
$$\angle Y_2 P_3 Y_1 = \angle Y_1 P_3 X_2 = 90^\circ$$
$$\implies \angle Z_2 P_3 Y_1 = \angle Y_2 P_3 Y_1 \implies Y_2, Z_2, P_3 \text{ are collinear}$$
$$\implies Y_2, Z_2, X_2 \text{ are collinear.}$$