Reply To: Maths
Who said that geometry was boring? Here are three interesting questions related to a triangle ABC. Enjoy!
First we assume that ABC is an equilateral triangle of side length 1.
QUESTION 1
A farmer, Kim, wishes to choose a point P inside ABC, and connect P to the three sides AB,BC,CA using three pieces of fencing. Where should Kim place P, and how much fencing will he need?
I am guessing Kim wants to do this as cheaply as possible? Harry
QUESTION 2
Kim the farmer (see Question 1) is now chosen to connect A,B,C to three points X,Y,Z (in that order), which are on the interiors of sides BC,CA,AB (in that order). He decides to do this using three paths. However, the paths cannot cross any of the sides of ABC, due to walls (or some other contraption that blocks travel) being present, and they cannot cross each other. Can he lay down the three paths?
[Just to make it clear: the three paths must be in the plane of triangle ABC, and take the form of continuous curves which do not touch any part of triangle ABC apart from their two endpoints. All three paths must be pairwise disjoint. They can arrive at A/B/C/X/Y/Z from the inside or the outside of ABC. OK? Good.]
Now we let ABC be any triangle. The only theorem you need (which is basically stolen from GCSE circle theorems) is:
Let W,X,Y,Z be a convex cyclic quadrilateral. (That is, let W,X,Y,Z lie on a circle in that order.) Then:
- Angles WXZ and WYZ are equal.
- Angles WXY and WZY add to 180°.
What isn’t taught at GCSE is that the converse of this is true also. So, for example:
If WXYZ is a convex (informally: nice) quadrilateral, and angles WXZ and WYZ are equal, then W,X,Y,Z lie on a circle in that order.
And here is the question!
QUESTION 3 (Harder!)
Let Γ be the circumcircle of ABC. (That is, let Γ be the (unique) circle which passes through all of A,B,C.) Consider the three circles Γ_A, Γ_B, Γ_C created by reflecting Γ over BC,CA,AB respectively.
Prove that Γ_A, Γ_B, Γ_C all concur at a point. (Extra marks if you can prove what that point is!)