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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 $\triangle ABC$. Extend the internal angle bisectors at $A$ and $C$ until they meet at a point $I$, and draw perpendiculars from $I$ to the three sides of $\triangle ABC$.

a. Using trigonometry or otherwise, show that these three perpendiculars all have the same length (call it $r$, the inradius).

b. Hence show that $\overline{IB}$ is also an angle bisector, and that these all meet at $I$ (the incentre).

c. A circle centered at $I$ and with radius $r$ (the incircle) 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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    $\begingroup$ 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. $\endgroup$ – Zev Chonoles Mar 2 '13 at 19:35
  • $\begingroup$ 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. $\endgroup$ – Zev Chonoles Mar 2 '13 at 19:36
  • $\begingroup$ Also, what school does your son go to where they learn about Gergonne points? Wish I had gone there... $\endgroup$ – Potato Mar 4 '13 at 7:17
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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.

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  • $\begingroup$ Thank you for your help @ Sarkar $\endgroup$ – Lucy Mar 2 '13 at 20:08

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