This is the Problem 6 of IMO 2018 https://www.imo-official.org/problems.aspx
I use the code at Mapleprimes (with some prepairs) https://www.mapleprimes.com/questions/228162-How-Can-I-Get-One-Option-Coordinates
a = {0, 0};
b = {5, 0};
c = {3, 4};
d = {15 - 4*Sqrt[19], 6};
x = {-172*Sqrt[19]*(1/757) + 1839/757, 2762/757 - 176*Sqrt[19]*(1/757)};
I checked the conditions and all of them are true.
EuclideanDistance[a, b] * EuclideanDistance[c, d] -
EuclideanDistance[a, d] * EuclideanDistance[c, b] // Simplify
VectorAngle[a - x, a - b] - VectorAngle[c - x, c - d] // Simplify
VectorAngle[b - x, b - c] - VectorAngle[d - x, d - a] // FullSimplify
But, when I change coordinates one point A, B, C, then sometimes, I can not get the result.
How can I always get one option of coordinates of the points A, B, C, D, and X of the Problem 6 in IMO 2018?