This booklet constitutes the refereed court cases of the 14th IAPR TC-18 foreign convention on Discrete Geometry for machine Imagery, DGCI 2008, held in Lyon, France, in April 2008. The 23 revised complete papers and 22 revised poster papers offered including three invited papers have been rigorously reviewed and chosen from seventy six submissions. The papers are prepared in topical sections on versions for discrete geometry, discrete and combinatorial topology, geometric transforms, discrete form illustration, reputation and research, discrete tomography, morphological research, discrete modelling and visualization, in addition to discrete and combinatorial instruments for photograph segmentation and research.

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ABC with AB = BC. To PROVE: LA = LC. 5 PROOF: Assume that BD, the bisector of L B, is drawn. ABD and CBD, -Given AB = BC - Identical BD= BD L ABD = L CBD - Each! s. :. &.. EXERCISES 1. Prove that the bisector of the vertex angle of an isosceles triangle is perpendicular to the base. 2. 6, La = L p, and AD = BE. ABE; and (b) list four other pairs of equal angles in the figure. 27 CONGRUENCE OF TRIANGLES c A~~--------~~B FIGURE 3. 6 AC and CD = CEo Prove that DA = BE. 4. 7. Show that for any angle of intersection of the bars, it is true that AB = CD, and AD = BC.

ABE; and (b) list four other pairs of equal angles in the figure. 27 CONGRUENCE OF TRIANGLES c A~~--------~~B FIGURE 3. 6 AC and CD = CEo Prove that DA = BE. 4. 7. Show that for any angle of intersection of the bars, it is true that AB = CD, and AD = BC. 7 5. 8, it is given that BD = DE = EC. Prove that LBAD = LCAE. CAD. 8 6. Find the hypotenuse c of a right triangle ,whose legs are given as indicated. Use the formula c = Va 2 + b2• (a) a =6",b =8". (d) a = 21",b =45". (b) a = 10", b = 24". , b = v'ls in.

With a radius larger than AB, describe arcs with B and C as centers and let these arcs intersect in a point called D. 16, is the required perpendicular. 16 BD = DC - Construction BA =AC - Construction AD=AD - Identical. s. Then ~BDA '" ~CDA Hence L BAD = L CAD - Corresponding parts of ~ & L BAD + L CAD = 1800 - Definition of a straight angle 0 L BAD = 90 Half of a straight angle, which establishes the construction. PROOF: C. To construct the bisector of a given angle. CONSTRUCTION: With the vertex A of the given L as the center and a convenient radius draw an arc cutting the two sides 11 and 12 of the given angle in points Band C, respectively.

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