I wanted to derive the formula to give the point of intersection of two lines, each defined by a pair of points. I got the wrong answer and cannot find the error. Which drives me crazy. I don't how how it could be more straightforward. I can get the answer off Wikipedia but I want to know what I did wrong.
I started by constraining the point of intersection $p$ to be colinear with both pairs ($a$ and $b$, $c$ and $d$).
Equating the slopes: $\dfrac{p_y - a_y}{p_x - a_x} = \dfrac{b_y - a_y}{b_x - a_x}$ and $\dfrac{p_y - c_y}{p_x - c_x} = \dfrac{d_y - c_y}{d_x - c_x}$
Solving for $p_y$: $p_y = \dfrac{b_y - a_y}{b_x - a_x}(p_x - a_x) + a_y$ and $p_y = \dfrac{d_y - c_y}{d_x - c_x}(p_x - c_x) + c_y$
Equating the two expressions and solving for $p_x$:
$\dfrac{b_y - a_y}{b_x - a_x}(p_x - a_x) + a_y = \dfrac{d_y - c_y}{d_x - c_x}(p_x - c_x) + c_y$ $p_x\left(\dfrac{b_y - a_y}{b_x - a_x}-\dfrac{d_y - c_y}{d_x - c_x}\right) = a_x\dfrac{b_y - a_y}{b_x - a_x} - a_y - c_x\dfrac{d_y - c_y}{d_x - c_x} + c_y$ $p_x = \dfrac{a_x\dfrac{b_y - a_y}{b_x - a_x} - a_y - c_x\dfrac{d_y - c_y}{d_x - c_x} + c_y}{\dfrac{b_y - a_y}{b_x - a_x}-\dfrac{d_y - c_y}{d_x - c_x}}$
This result for $p_x$ did not give me the expected values for the point of intersection in my graphing software. Wikipedia has:
$p_x = \dfrac{(a_xb_y - a_yb_x)(c_x - d_x) - (a_x - b_x)(c_xd_y - c_yd_x)}{(a_x - b_x)(c_y - d_y) - (a_y - b_y)(c_x - d_x)}$
Rearranged to highlight the differences:
$p_x = \dfrac{\dfrac{c_xd_y - c_yd_x}{d_x - c_x} - \dfrac{a_xb_y - a_yb_x}{b_x - a_x}}{\dfrac{b_y - a_y}{b_x - a_x} - \dfrac{d_y - c_y}{d_x - c_x}}$
While that expression was derived from determinants, I'd really appreciate knowing where I went wrong the way I was doing it.