Timeline for The locus of the intersection point of two perpendicular tangents to a given ellipse
Current License: CC BY-SA 3.0
9 events
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| Oct 3, 2011 at 17:48 | comment | added | Matt | To rederive this proof, I think it is sufficient to remember the following subgoals: 0. The proof makes heavy use of the "auxiliary circle" C, sharing a diameter with the ellipse. 1. Note that C is homothetic, from the point of view of one focus, to a circle twice its size centered at the other focus. 2. For any tangent T to the ellipse, the foci project perpendicularly onto T exactly where T intersects C. 3. Considering a point P as stated in the problem, the power (with respect to C) of P is the same (except for sign) as the power of either focus. Rather nice, actually. | |
| Apr 18, 2011 at 13:19 | vote | accept | Matt | ||
| May 4, 2014 at 10:29 | |||||
| Apr 18, 2011 at 13:17 | comment | added | Matt | For completeness, I will add the rest of the proof here: Let E,F be the foci of the ellipse, Q,S be their projections onto one tangent from A, and G,H be their projections onto the perpendicular tangent from A. Now the power of A with respect to the auxiliary circle (which the lemma above shows that G,H,Q,S are all on) is AQ AS = GE HF = [negative] power of F, which is a constant. Therefore the locus of A is a circle. | |
| Apr 18, 2011 at 10:48 | comment | added | Matt | Specifically, it proves that the projection of a focus onto a tangent will lie on the "auxiliary circle" (the circle having the ellipse's major axis as a diameter), like this: Let E, F be the foci, O the center, T the point of tangency, and Q the projection of E onto the tangent ("projection" meaning EQ ⊥ TQ). Extend EQ and FT to meet at R. It is a property of ellipses that ∠ETQ=∠RTQ (i.e., they reflect sound/light from one focus to the other), so ▵ETQ≅��RTQ. So QO is the midline of ▵REF, so 2 QO = RF = ET+TF = diameter of circle, making QO a radius of the circle. QED. | |
| Apr 18, 2011 at 9:51 | comment | added | Matt | Ah yes, page 174 (page 186 using their pdf reader) of the post-graduate thesis by Μπουνάκης, Δημήτρης in your link gives the answer. | |
| Apr 18, 2011 at 0:47 | comment | added | Henry | The name corresponds to a directrix for a parabola. I think the monograph referred to may be here but it is in Modern Greek and the pictures do not have circles [Adobe also suggests there is only 1 page, but there are up to 266]. | |
| Apr 18, 2011 at 0:37 | comment | added | J. M. ain't a mathematician | @Matt: you might sometimes see it referred to as the "Monge circle" as well. | |
| Apr 18, 2011 at 0:17 | comment | added | Matt | I didn't realize the circle has a name! The Pamfilos page is the sort of thing I was hoping for, but I'm missing something simple on the very first step: Why is it obvious that circle OP, circle GHG'H', and circle MNM'N' are all the same circle? (M,N,G, etc. being defined as projections of the foci onto the tangents.) The steps after that are fine. | |
| Apr 17, 2011 at 21:50 | history | answered | Henry | CC BY-SA 3.0 |