Exersise 01

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 (Bb Cc)/(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
• CA1A1Blm, BC1C1Anl, AB1B1Cmn
• Exersise 05
• Let A1, B1,C1 be the points on the sides of the
  triangle ABC.
• CA1A1B BC1C1A AB1B1C 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) ? (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,C1 be the points on the sides of the
  triangle ABC.
• Relation CA1A1B BC1C1A AB1B1C -1
• is a necessary and sufficient condition for the
  points
• A1, B1,C1 to be on the same line.

6
1.4 Exercises The 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 -gt X Mnn and
• translations X-gt X O .
• Prove that
• F (Aff (A1,, A2, , Ak))Aff (F A1, F A2, , F
  Ak).

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 PA1A2Ak , 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 ?k0 implies
?10,, ?k0 . Exercise 8 A set A1, A2,, Ak
is affinely independent if and only if the set of
vectors A1A2,, A1Akis linearly
independent.
10
fP

• Polarity fP preserves incidences. Prove that fP
  maps
• Exercise 9
• points of a plane ? to the planes through the
  point fP(?),
• Exercise 10
• points of a line x to the planes through a line
  (def.) fP(x),
• (an edge AB of a polyhedron to the edge fP(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 ? z0 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 MMM ? fP (A), A ?(A), equals
  d2 / 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 AiAk of the polygon
• P A1A2An divides the interior of P into the
  interiors of
• the polygons P1 AiAkAk1 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.