Circles \Gamma_1 and \Gamma_2 intersect at 2 distinct points A and B. A line l through A intersects \Gamma_1 and \Gamma_2 at C and D, respectively. Let M be the midpoint of CD. The line MB intersects \Gamma_1 and \Gamma_2 again at E and F , respectively. If MA=129, MB =156 and MC =182 , what is the value of EF ?
To find the value of EF, we can use the power of a point theorem.
The power of a point theorem states that if a line intersects a circle at two points, then the product of the lengths of the line segments from the points of intersection to the point outside the circle is equal. In equation form:
AC * AD = AE * AF
Let's denote the lengths of CE and DE as x and y, respectively. Since MC is the median, we can conclude that ME = MC/2 = 182/2 = 91.
Using the power of a point theorem, we can construct the following equation:
x * (x + 182) = (x + 273) * (x + y)
Simplifying this equation, we get:
x^2 + 182x = (x + 273) * (x + y)
Expanding the right side of the equation, we get:
x^2 + 182x = x^2 + (273x + 273y) + (x * y + 273y)
Combining like terms, we have:
182x = 273x + 273y + x * y + 273y
Rearranging the equation, we get:
91x = 546y + x * y
Now let's substitute the given values:
91 * 156 = 546y + 156y
14196 = 702y
Simplifying further:
y = 14196 / 702
y ≈ 20.249
Now substitute the value of y back into the equation:
91x = 546 * 20.249 + x * 20.249
Simplifying:
91x = 11018.754 + 20.249x
Combining like terms:
70.751x = 11018.754
Now solve for x:
x = 11018.754 / 70.751
x ≈ 155.796
Since E is the second intersection point of MB with Γ₁, we know that length EF is equal to CE + CD. Therefore:
EF = x + y
EF ≈ 155.796 + 20.249
EF ≈ 176.045
So, the value of EF is approximately 176.045.