Çözüldü Coordinate Geometry - Trigonometry - Two-Variable Equation - Pythagorean and Angle Bisector Theorems

For the parallelogram ABCD, the points A and B are on the positive side of the x-axis, and the point D is on the positive side of the y-axis, where O is the origin of the Cartesian plane, so that the angles ∡ADO and ∡BDA are equal, and if the line
3x + 2y = 12 passes through B and C, find the sum of the coordinates of the point C.

The line on [BC] intercepts the x-axis at (12 / 3, 0) that is B(4, 0)
n ∈ R^(+) and the line on [AD] is y = -3x / 2 + n, then A(2n / 3, 0) and D(0, n)
|AB| = |4 - 2n / 3| = | (12 - 2n) / 3 | units
c ∈ R^(+) and C(c, n)
|CD| = c
|AB| = |CD| ⇒ | (12 - 2n) / 3 | = c ⇒ 2n + 3c = 12....(I)
∡ADO = ∡BDA = θ
tan(θ) = (2n / 3) / n = 2 / 3....(II)
tan(2θ) = 4 / n....(III)
tan(2θ) = 2·tan(θ) / { 1 - [ tan(θ) ]^2 }....(IV)
Using the equations (III) and (IV) 4 / n = 2·tan(θ) / { 1 - [ tan(θ) ]^2 } and the value in (II),
2 / n = (2 / 3) / [ 1 - (2 / 3)^2 ] ⇒ n = 5 / 3....(V)
Substituting the value in (V) for its place in (I) 2·(5 / 3) + 3c = 12 ⇒ c = 26 / 9....(VI)
Summing the values in (V) and (VI) 5 / 3 + 26 / 9 = 41 / 9.

Note: Applying the Pythagorean Theorem, |BD| = (4^2 + n^2)^0.5 unit, and using the Angle Bisector Theorem on the BOD right triangle;
(2n / 3) / n = (4 - 2n / 3) / (4^2 + n^2)^0.5 ⇒ n = 5 / 3, then the rest is the same.