Let AB be the diameter of circle Γ1. In the interior of Γ1, there are circles Γ2 and Γ3 that are tangent to Γ1 at A and B, respectively. Γ2 and Γ3 are also externally tangent at the point C. This tangent line (at C) cuts Γ1 at P and Q, with PQ=20. The area that is within Γ1 but not in Γ2 or Γ3 is equal to Mπ. Determine M

To determine M, we need to find the area within Γ1 (circle with diameter AB) but not in Γ2 or Γ3.

First, let's observe that P and Q are points on the circumference of Γ1, which means the line through P and Q is a chord of Γ1.

Since PQ = 20, we know that the length of the chord is 20. Using the properties of a circle, we can determine the radius of Γ1 and the lengths of the other two radii.

Let's denote the center of Γ1 as O and let r be the radius of Γ1.

Since AB is the diameter, we have AB = 2r.

Since PAQ is a right triangle with the hypotenuse PQ, we can use the Pythagorean theorem:

(PQ/2)^2 = r^2 - (r - r2)^2

10^2 = r^2 - r^2 + 2rr2 - r2^2

100 = 2rr2 - r2^2
100 = r2(2r - r2)

Simplifying this equation, we have:

2r - r^2 = 50/r2

Now, let's consider the Area(Γ1) - Area(Γ2) - Area(Γ3) = Mπ.

The area of a circle is given by A = πr^2, so the area of Γ1 is πr^2.

To find the area of Γ2 and Γ3, we need to determine their radii first.

Γ2 is tangent to Γ1 at point A, so its radius is the length of the line segment AC. Since Γ2 and Γ3 are externally tangent at C, AC is also the line segment CD, which is the radius of Γ3.

To find AC and CD, we can use similar triangles.

The triangle ABC is similar to triangle CPQ, so we have:

AB/PQ = BC/CQ.

Substituting the given values, we get:

2r/20 = r - r2/CD
2r/20 = r(1 - r2/CD)

Simplifying, we have:

CD = 20(r/r - r2)

Now, we can find the area of Γ2 and Γ3 using the formulas:

Area(Γ2) = π(AC)^2
Area(Γ3) = π(CD)^2

Substituting the values we found earlier, the area outside Γ2 and Γ3 is:

Mπ = Area(Γ1) - Area(Γ2) - Area(Γ3)
= πr^2 - π(AC)^2 - π(CD)^2
= π(r^2 - AC^2 - CD^2)

By substituting the expressions we found for AC and CD, we can simplify further:

Mπ = π(r^2 - (r - r^2)^2 - 20(r/r - r^2)^2)

Expanding and simplifying the expression, we have:

Mπ = π(r^2 - (r^2 - 2rr^2 + r^4) - 20(1 - 2r + r^2)^2)
= π(r^2 - r^2 + 2rr^2 - r^4 - 20(1 - 2r + r^2)^2)

Expanding further, we have:

Mπ = π(2rr^2 - r^4 - 20(1 - 4r + 4r^2 - 2r^2 + 4r^3 - r^4))

Simplifying the expression, we get:

Mπ = π(2rr^2 - r^4 - 20 + 80r - 80r^2 + 40r^2 - 80r^3 + 20r^4)

Combining and rearranging like terms:

Mπ = π(20r^4 - 80r^3 + 40r^2 - 80r + 20 - r^4 + 2rr^2)

Mπ = π(19r^4 - 80r^3 + 2rr^2 + 40r^2 - 80r + 20)

Therefore, M = 19r^4 - 80r^3 + 2rr^2 + 40r^2 - 80r + 20.