Geometry Stumper

[Thomas Puettmann]

(08-14-2018 06:58 PM)Albert Chan Wrote:  What does midpoint of a circle mean ?

center, sorry.

A standard theorem from elementary geometry says that all three (interior) angular bisectors meet at one point, namely, the center of the inscribed circle.

How can one proof this? Imagine an arbitrary point on an angular bisector. Keeping this point as the center start blowing this point into a larger and larger circle. For one and only one radius the circle will touch both rays of the angle simultaneously.

Now, imagine you have two (interior) angular bisectors in a triangle. They meet at one and only one point. The circle just described now touches all three sides of the triangle. This circle is called inscribed circle of the triangle. It follows immediately that the third angular bisector must pass through the center of this circle.


To conclude the proof of your original problem you do the same at D outside of the triangle ABC. Just watch out what you get for free.

To those who like problems like this and would like to develop some geometric reasoning skills, I cannot recommend highly enough the book "Modern Geometry with Applications" by George A. Jennings. This is the book I really learned geometry from (actually quite late when I had to give my first course on the subject), and it is still one of my favourite books.