I have an ellipse of semi-axis major a along x-axis and semi-axis minor b along y-axis.
Having a point C defined with its geodetic (not geocentric) latitude $\varphi_1$ on the ellipse surface then drawing a point D along the normal line by height h, so segment CD = h.
Then I draw the tangent lines from D to the ellipse, they touch the ellipse at tangent points F and E respectively.
Another point G, also defined by its geodetic latitude $\varphi_2$.
The normal line at point G intersects the tangent line (DE) at point H.
My question is, how to find the height m of the segment [GH]?
For a circle I would know how to calculate it but for an ellipse, it's complicated :(
Illustration:
It is cumbersome algebraic work. But I shall indicate the method for you to complete.
At first solve together standard form and its derivative for coords ${C ,G. }$
Let $$ t1= \tan \varphi_1 \,; t2= \tan \varphi_2\,\tag1$$ $$ \frac{x_1^2}{a^2} +\frac{y_1^2}{b^2}=1;\, \frac{x_1}{a^2} +\frac{y_1 t2}{b^2}=0;\,\frac{x_2^2}{a^2} +\frac{y_2^2}{b^2}=t2 ;\, \frac{x_2}{a^2} +\frac{y_2 t2}{b^2}=0; \tag2 $$
to get points on ellipse $(C,G)$
$$x_1= \frac{a^2\, t1}{\sqrt{b^2+a^2t1}}, \,y_1= \frac{b^2}{\sqrt{b^2+a^2t1}}\,;\tag3$$
$$x_2= \frac{a^2\, t2}{\sqrt{b^2+a^2t2}}, \,y_2= \frac{b^2}{\sqrt{b^2+a^2t2}}\,;\tag4 $$
Next coordinatess of $D,H$ as outward extensions along each normal
$$ X_1=x_1 + h \cos \varphi_1; Y_1=y_1 + h \sin \varphi_1; \tag5$$
$$ X_2=x_2 +m \cos \varphi_2; Y_2=y_2 +m \sin \varphi_2; \tag6$$
Equation of line joining $DH$
$$ \dfrac{y-Y_1}{x-X_1}=\dfrac{Y_2-Y_1}{X_2-X_1} \tag7 $$
Put the above into $Ax+By+C=0 $ form.
First find condition that $Ax+By+ C=0$ has tangency w.r.t. the standard ellipse.
The discriminant $\Delta$ of
$$ \dfrac{x^2}{a^2} +\dfrac{1}{b^2}[(C+Ax)/B]^2 -1=0 \tag8$$
Or, ( check this up!)
$$ (CA/b^2B^2)^2 = (1/a^2 +A^2/b^2B^2) *[( C^2-b^2B^2)/b^2 B^2] \tag9$$
After checking the above take it forward by simplifying it to find the required formula/condition. It would be implicit in $m$ ..
Verify $m$ value found out on your Geogebra? graph with numerical values you gave in to make it.