Geometry Stumper

[Thomas Puettmann]

(08-16-2018 12:30 AM)jwhsu Wrote:  Is this basically using an excircle to solve the problem?

Yes, I forgot about the name. In mathematics you try to memorize as little as possible. Here, this means essentially just the reflex "angular bisectors" --> "inscribed circle". The definition, the construction and the properties of the three excircles are so analogous to that of the inscribed circle, that you simply recover them when needed.

(08-15-2018 11:02 PM)Albert Chan Wrote:  Your proof should belong in the Book.

Euclid and his ancestors have shown how to use inscribed circles and excircles and their proofs definitely belong in the Book. "Seeing" in elementary geometry is a lot of fun, but there is almost no way to do something original. It's an art of recreation. All simple problems like this have been solved thousands if not millions of times with minor variations.

With the notion of an excircle, here goes the complete proof.

- The angular bisector of the interior angle at B and the angular bisector of the exterior angle at A meet at the center D of one of the three excircles of the triangle ABC.
- This excircle touches the three lines AB, AC, and BC.
- The line DC is the angular bisector of the exterior angle at C.

- By the symmetric construction on the other side, the line EC is also the angular bisector of the exterior angle(s) at C.
- Since the angular bisector of the exterior angles at C is unique, we have that the line DC is equal to the line EC.

No need to produce an 180° angle at C.