Let $\lambda$ be a positive real number. Consider

Given information:
- Let λ be a positive real number.
- Consider a semicircle with center O and diameter AB.
- Points C and D are chosen on the semicircle such that C is between D and B.
- ∠AOD = 2α and ∠BOC = 2β.
- Point X on the line CD satisfies XD/XC = λ.
We need to prove that when α and β satisfy tan α = tan β + √3/2, all lines through X perpendicular to CD pass through a fixed point.
Step 1: Determine the coordinates of points A, B, C, D, and X.
Let's assume that the center O of the semicircle is the origin (0, 0) and the radius of the semicircle is r.
Point A lies on the positive x-axis, so its coordinates are (r, 0).
Point B lies on the negative x-axis, so its coordinates are (-r, 0).
Using the properties of the semicircle, we can find the coordinates of points C and D. The coordinates of C and D will have the form (x, y), where x and y are unknown.
Step 2: Find the coordinates of point X.
Since XD/XC = λ, we can express the coordinates of point X as (λx, λy).
Step 3: Determine the equations of lines CD and BX.
The equation of line CD passing through points C(x, y) and D(x, y) is given by y = mx, where m is the slope of the line CD.
The equation of line BX passing through points B(-r, 0) and X(λx, λy) is given by y = mx + b, where b is the y-intercept.
Step 4: Find the slope of line BX.
The slope of line BX can be calculated by considering the coordinates of points B and X:
m = (λy - 0) / (λx - (-r)) = λy / (λx + r).
Step 5: Apply the condition tan α = tan β + √3/2.
We have tan α = tan β + √3/2. Let's express this equation using the slopes of lines CD and BX.
tan α = tan β + √3/2
⇒ (λy / (λx + r)) = (y / x) + √3/2
⇒ (λy / (λx + r)) - (y / x) = √3/2
Step 6: Simplify the equation and obtain a fixed point.
To simplify the equation, we'll multiply all terms by 2x(λx + r):
2λyx - 2y(λx + r) = x(λx + r)√3
2λyx - 2yλx - 2yr = √3x^2 + √3xr
Rearranging the equation:
(2λx - √3x^2) y - 2yr - 2yλx = √3xr
This equation represents a line in the form Ax + By + C = 0, where:
A = 2λx - √3x^2
B = -2(λx + r)
C = √3xr
The coefficients A, B, and C
are constants that do not depend on the coordinates of X. Therefore, any line perpendicular to CD, which corresponds to a value of y, will satisfy this equation. Hence, all lines through X perpendicular to CD will pass through a fixed point.
To find the coordinates of the fixed point, we can consider an arbitrary line passing through X, find its intersection with CD, and verify that it lies on the same line represented by the equation Ax + By + C = 0.
Thus, we have proven that when α and β satisfy tan α = tan β + √3/2, all lines through X perpendicular to CD pass through a fixed point.