# Trigonometry and Geometry

I have no idea on how to solve this question so can someone please assist me. My son brought it from school and he is really struggling with the question.
Consider a triangle ABC with line segments cutting the internal angles at A and C in half (the angle bisectors). Extend these lines until they meet at a point I and draw perpendiculars from I to the three sides of ABC. (a) Using trigonometry or otherwise, show that these three perpendiculars all have the same length (call it r, the in radius). (b) Hence show that the line segment IB is also an angle bisector and that these all meet at I (the in centre). (c) A circle centred at I and with radius r (the in circle) is tangent to each side of the triangle. Show that the lines joining the points of contact to the opposite vertices all meet at a single point (called the Gergonne point).

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Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself. Also, some would consider your post rude because it uses commands ("Show"), not a request for help, so please consider rewriting it. – Zev Chonoles Mar 2 '13 at 19:35
Also, you can find some good starting points on how to format mathematics on the site here. This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.SE site. – Zev Chonoles Mar 2 '13 at 19:36
@ZevChonoles he probably used the imperative case because it was in the question; one of the tags is 'Homework' so that is probably how the homework question is phrased – imulsion Mar 2 '13 at 19:46
@Zev, it is hard to believe that a parent who probably never heard of TeX should learn it in order to ask a few questions about kids' homework assignments. Also, are you seriously proposing that after saying "this is my son's homework assignment" it is rude to present the complete text of the question, including imperatives? – zyx Mar 2 '13 at 20:25
@zyx: You apparently didn't notice that the part about the OP being a parent was added after I posted my comments. I also find it quite reasonable to preemptively mention to people (whether or not they are a parent or student) where to learn how to use the basics of MathJax, which is trivial compared to the whole of TeX, even if their current question does not require it, if I think they may ask future questions which would benefit from it. – Zev Chonoles Mar 2 '13 at 20:36

This is the best diagram I found on the internet in image search (next time, try to provide a diagram with the question). It calls the point inside $O$ instead of $I$. Let the point on $AC, CB, BA$ be $X, Y, Z$ respectively. Then all you have to show for the first part is that triangle $AOX$ is congruent to triangle $AOZ$. This is easy because they are both right angled, have equal half angles at $A$ and share a common side $AO$. So $OX = OZ$. You can prove that $OZ = OY$ with a similar argument and so all perpendiculars have the same length. For the next part, you can show that triangle $BOZ$ is congruent to $BOY$. So the two angles at $B$ are equal and hence $OB$ is an angle bisector. A tangent to a circle at point $P$ is perpendicular to $OP$ and passes through $P$. All the sides satisfies this, so they are tangent to the incircle. I am not sure how to do the Gergonne point question.