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Exercise sheet 3
Exercise 1.We start with a triangleABC, for instanceA.2; 4/,B.0; 0/and C.6; 0/. Let PQR be the orthic triangle of the triangle ABC, with P on BC,QonCAandRonAB. LetK be the intersection ofBC with the line throughAwhich is parallel toRP.
1°) Draw an exact picture of the trianglesABC andPQR.
2°) What are the coordinates of the pointK?
3°) Draw a trajectory starting at some pointM on the segmentPK bet- weenP andK in a direction parallel toPQ.
4°) What happens if you start fromK in the direction ofPQ? 5°) What happens if you start from.72; 0/in the direction ofPQ?
B
A
C R
Q
P
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K
Exercise 2.LetC be a circle withe centerOand radius5. LetC0be another circle with same center O but with a smaller radius 5
p2
2 . We consider the billiard whose border is the circleC. Find a trajectory which is tangent to C0. Is this trajectory periodic ? what do you think would happen if we take C0to be the circle with radius3;54?
O C’
C
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