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01.4 Exercises The excersises 01-011are not obligatory. They are due in 2 weeks.

01.4 Exercises The excersises 01-011are not obligatory. They are due in 2 weeks. 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 = ( B • b + C • c )/( b + c ) Exersise 02

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01.4 Exercises The excersises 01-011are not obligatory. They are due in 2 weeks.

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  1. 01.4 ExercisesThe excersises 01-011are not obligatory. They are due in 2 weeks. Exersise 01 If A1 is the foot of the bisector at A of the triangle ABC, and b, c the lengths of the sides CA, AB respectively,then A1 = (B•b + C•c)/(b + c) Exersise 02 If A1 is the foot of the normal at A of the triangle ABC, then A1 = (B• tan + C• tan)/(tan + tan) Exersise 03 If the line AA1, A1 BC contains the center O of the cirumscribed circle of ABC, then A1 = (B• sin2 + C• sin2)/(sin2 + sin2)

  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 CA1:A1B=l:m, BC1:C1A=n:l, AB1:B1C=m:n Exersise 05 Let A1, B1,C1be the points on the sides of the triangle ABC. CA1:A1B * BC1:C1A * AB1:B1C = 1 is a necessary and sufficient condition for the lines AA1,BB1,CC1 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 A1 = (A tan + B tan + C tan)/(tan + tan + tan) Exersise 09 The center of the circumscribed circle of the triangle ABC satisfies A1 = (A sin2 + B sin2 + C sin2)/(sin2 + sin2 + sin2)

  4. Exersise 010 If the points M and N have affine coordinates (m1,m2,m3) and (n1,n2,n3) wrt some points A,B,C, then the points X of the line MN have the affine coordinates (x1,x2,x3)= μ (m1,m2,m3) + γ (n1,n2,n3) wrt A,B,C. Proof that, up to a scalar multiple, there exists a unique triplet (p1,p2,p3) of real numbers having the property: (p1,p2,p3) * (x1,x2,x3) = 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 A1, B1,C1be the points on the sides of the triangle ABC. Relation CA1:A1B * BC1:C1A * AB1:B1C = -1 is a necessary and sufficient condition for the points A1, B1,C1 to be on the same line.

  6. 1.4 ExercisesThe exercises 1-7 are due in 2 weeks Exercise 1 Prove that if a point B belongs to the affine hull Aff (A1, A2, …, Ak ) of points A1, A2,…, Ak , then: Aff (A1, A2,…, Ak) = Aff (B,A1, A2,…, Ak).

  7. Exercise 2 Prove that the affine hull Aff (A1, A2, …, Ak ) of points A1,A2,…, Ak contains the line AB with each pair of its points A,B. Moreover, prove that Aff (A1, A2, …, Ak) is the smallest set containing {A1, A2, …, Ak} and having this property. (Hint: proof by induction.) Exercise 3 Affine transformations F of coordinates are matrix multiplications : X -> X• Mnn and translations: X-> X + O’ . Prove that F (Aff (A1,, A2, …, Ak))=Aff (FA1, F A2, …, FAk).

  8. Exercises 1’- 3’ Reformulate exercises 1-3 by substituting affine hulls Aff (A1, A2, …, Ak) with convex hulls Conv (A1, A2, …, Ak), lines AB with segments [A,B]. Prove that: Exercise 4 If a convex set S contains the vertices A1, A2, …, Ak of a polygon P=A1A2…Ak , 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 (A1, A2, …, Ak) = Aff (A1, Aff (A2, …, Ak )). Conv (A1, A2, …, Ak) = Conv (A1, Conv (A2, …, Ak)). Exercise 7 A set {A1, A2,…, Ak} is affinely independent if and only if 1A1 +…+kAk =0, 1+…+k=0 implies 1=0,…,k=0 . Exercise 8 A set {A1, A2,…, Ak} is affinely independent if and only if the set of vectors {A1A2,…, A1Ak}is linearly independent.

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

  11. Let P be the palaboloid X2 + Y2 = 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 = r2 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 fP(A), A= (A), equals d2 / 2. ~

  12. Ecersise 014 (0... arenot obligatory) Prove that every polygon P has an inner diagonal (a diagonal having only inner points). Ecersise 015 Prove that an inner diagonal AiAk of the polygon P = A1A2…An divides the interior of P into the interiors of the polygons P1 = AiAkAk+1… and P2 = AiAkAk-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.

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