# Angle Bisectors in Ellipse II

Let A and B be two points on an ellipse with foci E and F. The tangents to the ellipse at A and B meet in S. Prove that

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Copyright © 1996-2018Alexander Bogomolny

Let A and B be two points on an ellipse with foci E and F. The tangents to the ellipse at A and B meet in S. Prove that

### Proof

To see why this is so, draw an ellipse through S *confocal* to the given one. This reminds of the configuration which showed the ellipse as an envelope of a family of straight lines.

So we now have two confocal ellipses: *ellipse*_{1} (the original one) and *ellipse*_{2} (passing through S.) The configuration admits the following interpretation. Draw a tangent to *ellipse*_{1} at A till it hits *ellipse*_{2} in S. Reflect it at S and continue to the next intersection with *ellipse*_{2}, and so on. All so constructed lines will touch an ellipse confocal with *ellipse*_{2} which is bound to be *ellipse*_{1} since the latter, by the construction, is already tangent to the first line AS. It follows that the second line in the chain is necessarily BS, implying that AS and BS are equally inclined to the tangent to *ellipse*_{2} at S. But so are ES and FS, and we are done.

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Copyright © 1996-2018Alexander Bogomolny