Let the ellipse be:
$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$
Set $P(a,t)$, the tangent line $PT_1$ is $y=k(x-a)+t$, $k=\tan A_1$, since it is tangent line, so put in ellipse equation, we have:
$$ b^2x^2+a^2(kx-ka+t)^2=a^2b^2$$
i.e.,
$$(b^2+a^2k^2)x^2+2a^2k(t-ka)x+a^2(t-ka)^2-a^2b^2=0$$
Solving for the point of tangency, the discriminant $\Delta$ must be zero:
$$\Delta$=$4a^4k^2(t-ka)^2-4(b^2+a^2k^2)[a^2(t-ka)^2-a^2b^2] $$
At $\Delta=0$, we get $ 2kta-t^2+b^2=0$, that is, $k=\dfrac{t^2-b^2}{2ta}$.
Let $k_1$ be $PF_1$'s slope, $k_1=\dfrac{t}{a+c}=\tan A_2$, the $ \angle T_1PF_1=|A_1-A_2|$, now we calculate $\tan(A_1-A_2)$:
$$\tan(A_1-A_2)=\dfrac{\tan A_1-\tan A_2}{1+\tan A_1 \tan A_2}=\dfrac{k-k_1}{1+k k_1}=\dfrac{\dfrac{t^2-b^2}{2ta}-\dfrac{t}{a+c}}{1+\dfrac{t^2-b^2}{2ta}\dfrac{t}{a+c}}=\dfrac{(t^2-b^2)(a+c)-2at^2}{2at^2+2act+t^3-tb^2}$$
Since $b^2=a^2-c^2$, we get:
$$ RHS=\dfrac{(t^2-a^2+c^2)(a+c)-2at^2}{t(2a^2+2ac+t^2-a^2+c^2)}=\dfrac{c-a}{t}$$
Let $k_2$ be line $PF_2$'s slope, then $k_2=\dfrac{t}{a-c}=\tan\angle PF_2T_2$. Since $\angle T_2PF_2=\dfrac{\pi}{2}-\angle PF_2T_2$, so $\tan \angle T_2PF_2=\dfrac{1}{\tan\angle PF_2T_2}=\dfrac{a-c}{t}=\tan(A_2-A_1)$. Since the angles are acute, we have:
$\angle T_2PF_2=A_2-A_1=\angle T_1PF_1$
$\mathrm{Fig.\space 1}$ shows what you have done up to now. The given information is shown in $\color{red}{\text{red}}$. The black lines and points are what you have constructed. By the way, we renamed your axisymmetric image of point $P$ as $Q$.
$\mathrm{Fig.\space 2}$ shows what you should do to locate the vertex $A$ of $\triangle ABC$. Mark two arbitrary points $M$ and $N$ on the given line $t$ and draw the two rays $PM$ and $PN$. Now locate the point $K$ on $PM$ such that $MK=2PM$. Similarly, mark the point $L$ on $PN$ such that $LN=2PN$. To complete the construction join the points $K$ and $L$ to intersect the line $BQ$ as shown in the diagram. The point of intersection of $BQ$ and $KL$ is the sought vertex $A$.
I think that you are versed in geometry to figure out why the steps described above lead to what you were looking for.
Best Answer
(interactive version here : moving the blue point gives a variable tangent ; changing the value of $b$ controls the eccentricity of the ellipse ; the choice has been made to keep $a=1$ fixed, giving $f=\sqrt{1-b^2}$). The radius of the circle is $2a=2$.
Three steps :
The set of symmetrical (or "mirror") points of focus $F'$ with respect to all tangent lines is known to be a circle centered in the other focus $F$.
Therefore, a fact you have remarked, having two of these mirror points, $G_1$ and $G_2$, $G_1G_2$ is a chord of this circle ; as a consequence, the bissector line of this chord passes through the circle's center $F$.
Tangency point $D_1$ being known, the reflection property of the ellipse implies that line $F_1D_1$ passes through focus $F$.
Conclusion : $F$ is defined as the intersection of line $F_1D_1$ with the bissector line of $[G_1G_2]$.