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Exercise sheet 3

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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

˛

˛

ˇ ˇ

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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