Geometric construction of the fourth intersection points of two conics

Based on Projective conic sections - constructions, the crux of the construction is this:

• let two conics intersect in $A,B,C,D$.
• let any line through $A$ intersect the conics again in $M,M'$
• let any line through $B$ intersect the conics again in $N,N'$
• then $MN, M'N'$ and $CD$ are concurrent.

To show this, consider the hexagons $MACDBN and M'ACDBN'.$ Let $P=MA\cdot DB=M'A\cdot DB$ and $Q=AC\cdot BN=AC\cdot BN'$. By Pascal's Theorem $CD\cdot MN$ and $CD\cdot M'N'$ are on line $PQ$, and the concurrency follows. In particular, $F=MN\cdot M'N'$ lies on $CD.$

For any $T\neq U$, let $UT\cdot UVWXY$ denote the other intersection $Z$ of the line $UT$ with the conic defined by the five points $U,V,W,X,Y$. There is a classic straightedge construction of $Z$ based on Pascal's Theorem which is described in Hatton's Projective Geometry (pg 240, 133.A.ii)

Putting it all together, the steps for the construction of $D$ are:

• $M=AT\cdot ABCD_1E_1$
• $M'=AT\cdot ABCD_2E_2$
• $N=BT\cdot BACD_1E_1$
• $N'=BT\cdot BACD_2E_2$
• $F=MN\cdot M'N'$ (as mentioned above, $F$ will lie on $CD$)
• $D=CF\cdot CABD_1E_1$

Note that the construction can be done with a straightedge only - no compass required!

In affine coordinates where $A=(a,0)$, $B=(0,b)$, $C=(0,0)$, the two conics have the equations $$p_1(x^2-ax)+q_1(y^2-by)+r_1xy=0$$ $$p_2(x^2-ax)+q_2(y^2-by)+r_2xy=0$$ So the fourth point of intersection $F$ satisfies $$\frac{y}{x-a}=\frac{p_1 q_2-p_2 q_1}{q_1 r_2-q_2 r_1}$$ $$\frac{y-b}{x}=\frac{r_1 p_2-r_2 p_1}{p_1 q_2-p_2 q_1}$$ The left sides are naturally interpreted in terms of the angles $FAC$ and $FBC$, so this might lead to a nice construction.

Update: Using Cramer's rule to solve for the $p$'s, $q$'s, $r$'s, we can take $$p_1=\begin{vmatrix} D_{1y}^{\,2}-bD_{1y} & D_{1x}D_{1y}\\ E_{1y}^{\,2}-bE_{1y} & E_{1x}E_{1y}\\ \end{vmatrix}$$ and similarly for $q_1, r_1, p_2, q_2, r_2$. So translating all this into geometry seems like it will primarily involve ten constructions of determinant calculations.