1. Exersise 01 If A 1 is the foot of the bisector at A of the triangle ABC, and b, c the lengths of the sides CA, AB respectively, then A 1 = (Bb + Cc)/(b + c) Exersise 02 If A 1 is the foot of the normal at A of the triangle ABC, then A 1 = (B tan  + C tan  )/(tan  + tan  ) Exersise 03 If the line AA 1, A 1  BC contains the center O of the cirumscribed circle of ABC, then A 1 = (B sin2  + C sin2  )/(sin2  + sin2  ) 01.4 Exercises The excersises 01-011are not obligatory. They are due in 2 weeks.

2. Exersise 04 If (l,m,n) are an affine coordinates of X wrt. A,B,C i.e. if (l + m + n)X = lB + mC + nA, then the lines AX, BX, CX intersect the opposite sides of the triangle ABC in points A,B,C dividing this segments in ratios CA 1 :A 1 B=l:m, BC 1 :C 1 A=n:l, AB 1 :B 1 C=m:n Exersise 05 Let A 1, B 1,C 1 be the points on the sides of the triangle ABC. CA 1 :A 1 B * BC 1 :C 1 A * AB 1 :B 1 C = 1 is a necessary and sufficient condition for the lines AA 1,BB 1,CC 1 to be in the same bundle (intersect or parallel to each other).

3. Exersise 06 The baricenter T of the triangle ABC satisfies T = (A + B + C)/3 Exersise 07 The center S of the inscribed cicrle in the triangle ABC satisfies S = (Aa + Bb + Cc)/(a + b + c) Exersise 08 The orthocenter of the triangle ABC satisfies A 1 = (A tan  + B tan  + C tan  )/(tan  + tan  + tan  ) Exersise 09 The center of the circumscribed circle of the triangle ABC satisfies A 1 = (A sin2  + B sin2  + C sin2  )/(sin2  + sin2  + sin2  )

4. Exersise 010 If the points M and N have affine coordinates (m 1,m 2,m 3 ) and (n 1,n 2,n 3 ) wrt some points A,B,C, then the points X of the line MN have the affine coordinates (x 1,x 2,x 3 ) = μ (m 1,m 2,m 3 ) + γ (n 1,n 2,n 3 ) wrt A,B,C. Proof that, up to a scalar multiple, there exists a unique triplet (p 1,p 2,p 3 ) of real numbers having the property: (p 1,p 2,p 3 ) * (x 1,x 2,x 3 ) = 0 for every X  MN. Such triplets are called affine coordinates of MN wrt A,B,C. Exersise 011 The afine coordinates of each point have a representative with the sum of coordinates 1. Prove that the affine coordinates of lines have the same property.

5. Exersise 012 Let A 1, B 1,C 1 be the points on the sides of the triangle ABC. Relation CA 1 :A 1 B * BC 1 :C 1 A * AB 1 :B 1 C = -1 is a necessary and sufficient condition for the points A 1, B 1,C 1 to be on the same line.

6. Exercise 1 Prove that if a point B belongs to the affine hull Aff (A 1, A 2, …, A k ) of points A 1, A 2,…, A k, then: Aff (A 1, A 2,…, A k ) = Aff (B,A 1, A 2,…, A k ). 1.4 Exercises The exercises 1-7 are due in 2 weeks

7. Exercise 2 Prove that the affine hull Aff (A 1, A 2, …, A k ) of points A 1,A 2,…, A k contains the line AB with each pair of its points A,B. Moreover, prove that Aff (A 1, A 2, …, A k ) is the smallest set containing {A 1, A 2, …, A k } and having this property. (Hint: proof by induction.) Exercise 3 Affine transformations F of coordinates are matrix multiplications : X -> X M nn and translations: X-> X + O’. Prove that F (Aff (A 1,, A 2, …, A k ))=Aff (F A 1, F A 2, …, F A k ).

8. Exercises 1’- 3’ Reformulate exercises 1-3 by substituting affine hulls Aff (A 1, A 2, …, A k ) with convex hulls Conv (A 1, A 2, …, A k ), lines AB with segments [A,B]. Prove that: Exercise 4 If a convex set S contains the vertices A 1, A 2, …, A k of a polygon P=A 1 A 2 …A k, it contains the polygon P. (Hint: Interior point property). Exercise 5-5’ Aff (S) (Conv ( S)) is the smallest affine (convex) set containing S, i. e. the smallest set X which contains the line AB (segment [AB]) with each pair of points A, B  X.

9. Prove that: Exercise 6-6’ Aff (A 1, A 2, …, A k ) = Aff (A 1, Aff (A 2, …, A k )). Conv (A 1, A 2, …, A k ) = Conv (A 1, Conv (A 2, …, A k )). Exercise 7 A set {A 1, A 2,…, A k } is affinely independent if and only if 1 A 1 +…+ k A k =0, 1 +…+ k =0 implies 1 =0,…, k =0. Exercise 8 A set {A 1, A 2,…, A k } is affinely independent if and only if the set of vectors {A 1 A 2,…, A 1 A k }is linearly independent.

10. Polarity f P preserves incidences. Prove that f P maps : Exercise 9 points of a plane  to the planes through the point f P (  ), Exercise 10 points of a line x to the planes through a line (def.) f P (x), (an edge AB of a polyhedron to the edge f P (A)  f P (B)) Exercise 11 points of the paraboloid P to the planes tangent to P Ecersise 013 (not obligatory) point A to the plane f P ( A ) containing the touching points of the tangent lines from A onto the paraboloid P. fPfP

11. Let P be the palaboloid X 2 + Y 2 = 2z and let  be the projection of the plane  : z=0 onto . Prove that: Exercise 12a The points of a circle k: (x-a) 2 + (x-b) 2 = r 2 in  are mapped (projected by  ) to the points of some plane . Exercise 12b The image of an interior point of k is below  =  (k). Exercise 13 If the distance d(M,A) equals d, than the distance between M=  (M) and M=MM  f P (A), A=  (A), equals d 2 / 2. ~

12. Ecersise 014 (0... are not obligatory) Prove that every polygon P has an inner diagonal (a diagonal having only inner points). Ecersise 015 Prove that an inner diagonal A i A k of the polygon P = A 1 A 2 …A n divides the interior of P into the interiors of the polygons P 1 = A i A k A k+1 … and P 2 = A i A k A k-1 … Ecersise 016 Prove that the vertices of any triangulation of a polygon P can be colored in 3 colors so that the vertices of every triangle have different colors.

13. Ecersise 017 Let S be a finite set of points in the plane and D its Delaunay Graph. Prove that a subset of S defines a facet of D iff the points of S lay on a circle which contains (on and in it) no other point of S. Ecersise 018 Prove that the greedy algorithm “edge flipping” leads to the Delaunay triangulation.