A pair of fixed parallel tangents on 2 points (R, R' respectively) of an ellipse are intercepted by a tangent generated from a random point P on the same ellipse. This random tangent meet parallel tangent of R at T, and met tangent of R' at T' respectively. I need to prove that RT : R'T' = PT : PT'
I tried to find the coordinates of T and T' in standard form (intersection of y = mx +- (a^2m^2 b^2)^1/2 with tangent at P, namely xx'/a^2 + yy'/b^2 = 1).then I try to find distance of each RT, R'T', PT, PT' respectively but the variables are just too much to simplify) I tried to find coordinates in parametric form also results in very complicated expression. I know that R, R'are connected by a diameter of the ellipse but I'm not able to find similarly in the section required to prove RT : RT' = PT : PT'. Is this some theorem of ellipse? How can I prove this without actually compute the length of each line?
Let $a$ and $b$ be the lengths of the semi-major and semi-minor axis respectively, and let $y$ be line containing semi-minor axis. Consider dilation of the plane relative to $y$ with coefficient $\frac{b}{a}$. It is an affine transformation, it turns ellipse into a circle, tangents to ellipse map to tangents to circle. (Note also that it keeps lines, it keeps relations to be parallel and to be collinear.) If we denote by $P_0,R_0,R_0',T_0,T_0'$ the images of $P,R,R',T,T'$ respectively, note that $R_0T_0:R_0'T_0'= RT:R'T'$ because $RT\parallel R'T'$, and $P_0T_0:P_0T_0'=PT:PT'$ because $P,T,T'$ are collinear. Now you only have to solve the problem on the circle, i.e. to prove $R_0T_0:R_0'T_0'= P_0T_0:P_0T_0'$, but this follows by $T_0R_0=T_0P_0$ and $T_0'P_0=T_0'R_0'$.