Get a point on a line segment parallel to another line segment having common intersection line

Question

In the image above, coordinates of points $A,\, B,\, C,\, D$ and $E$, are known. Also line $AB$ is parallel to $EX_1$ and so are lines $AC$ and $EX_2$. How can I find coordinates of points $X_1$ and $X_2$.

Answer 1

Let $r = \frac {DE}{DA}= \frac {DX_2}{DC}= \frac {DX_1}{DB}$.

Then $X_1$ and $X_2$ are $r$ of the distance between $D$ and $B$ and between $D$ and $C$.

So $X_1= (D_x + r(B_x - D_x), D_y + r(B_y- D_y))$ and $X_2 =(D_x + r(C_x - D_x), D_y + r(C_y- D_y))$.

Answer 2

Hint: You can find equations of the lines $AB$ and $BD$. Then you find the line parallel to $AB$ and passes through $E$. Finally, you find intersection of the lines $BD$ and $EX_1$.

Answer 3

Working in homogeneous coordinates, $$X_1\sim (B\times D)\times((A-B)\times E) \\ X_2\sim (C\times D)\times((A-C)\times E).$$ This is basically Farrukh Ataev’s answer computed directly.