Circles Γ1 and Γ2 have centers X and Y respectively. They intersect at points A and B, such that angle XAY is obtuse. The line AX intersects Γ2 again at P, and the line AY intersects Γ1 again at Q. Lines PQ and XY intersect at G, such that Q lies on line segment GP. If GQ=255, GP=266 and GX=190, what is the length of XY?

To find the length of XY, we need to apply some geometry principles and use the given information.

Let's break down the problem step by step:

1. Since circles Γ1 and Γ2 intersect at points A and B, we can deduce that A and B lie on the radical axis of the circles, which is the line XY.

2. We are given that the angle XAY is obtuse. Since AX and AY are line segments that intersect the circles at points P and Q respectively, we can conclude that the triangle APQ is an obtuse triangle.

3. Since Q lies on line segment GP, we can use the angle-bisector theorem to determine the ratio between GQ and GP. According to the theorem, the ratio of GQ to GP is equal to the ratio of the distances from Q to X and Y, respectively. In other words, GQ/GP = QX/QY.

4. Given that GQ = 255 and GP = 266, we can substitute these values into the ratio: 255/266 = QX/QY.

5. From the previous step, we know that QX = GX (since G and X are the same point), so we substitute GX = 190: 255/266 = 190/QY.

6. Rearranging the equation, we find QY = 266 * 190 / 255.

7. Now, let's focus on finding QY and QX. Since Q is the point of intersection between the circle Γ1 and line AY, we can use the power of a point theorem. According to the theorem, the product of the distances from Q to A and Q to Y is equal to the square of the distance from Q to the center of Γ1.

8. We are given that X is the center of circle Γ1, and the distance from Q to X is GX = 190. Therefore, we have QX * QY = AX * XY, where AX is the radius of Γ1 and XY is the length we want to find.

9. From the previous step, we know QX = GX = 190.

10. Substituting the known values, we have 190 * QY = AX * XY.

11. We don't know the value of AX (the radius of circle Γ1), but we can express it in terms of XY. Since A and B lie on the radical axis XY, AB is perpendicular to XY. Therefore, the triangle AXB is a right triangle.

12. Using the Pythagorean theorem, we have AX^2 + BX^2 = (2 * AX)^2 = XY^2, where BX = AX (since the center of Γ2, Y, lies on AB).

13. Simplifying, we have 4 * AX^2 = XY^2.

14. Substituting the expression for AX in the equation derived in step 10, we have 4 * QY * XY = XY^2.

15. Canceling the XY on both sides of the equation, we obtain 4 * QY = XY.

16. Plugging in the previously determined value for QY (QY = 266 * 190 / 255), we find XY = 4 * 190 * 266 / 255.

17. Simplifying the expression, we get XY = 396.549.

So, the length of XY is approximately 396.549.