Dardanos Theorem revised

ώ ό:

α ά α ος α αθ ω έ ο ω ίο . Α. σιά α, Η ο ό ο -Μ α ο ό ο Μ α ι ού, ΜΠ Α οφοί ο ο C.I.C.A, Sophia Antipolis, Nice, France

23.04.17

ίωσ : ο Θ ώ α αι α ό ι ή ο ίθ αι αι σ Α Note: The theorem and its proof are also presented in English.

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ι ή.

Α ι ί ο ο ω ή α ος ύ ο ( , R), α βά ο αι ύο αί ς ο ές αι οβά αι ία ί ς ά ς. Κα ό ι οβά ο αι α έσα ύο ο ώ , ο άθ έ α ί ς ά ς ο ής. Θα α ο ι θ ί ό ι οβο ή ο έσο ς άθ ο ής σ ά , ί αι ο έσο ς οβο ής ς ιας ο ής ί ά .

ο

ώ

α ά α ος

α έσα M1, M3 ύο ο ής ί ς ά ς.

αί

ο

ώ

ύ ο ( , R)

οβά ο αι σ α έσα

οβο ώ

ς ιας

9

1 5 12

Α

Π1 1

Π2

2 1 2

1

7

3

4

Α ό ι ύ βο α: =// ίσο αι α ά ο, // α ά ο ^ ία , => α ά σ έ ια ώ αΑά : ό ί ο Α , α έσα ύο α α ι έ ώ ο , οβά ο αι σ α έσα οβο ώ ώ , ς ιας ί ς ά ς. 1. Έσ οι ο ές Α , ο ύ ο ( , R) αι 1 4 οβο ή ς Α σ αι 9 12 οβο ή ς σ Α , 1, 3, α έσα Α , , α ίσ οι α αι 7, 5 α έσα 1 4 αι 9 12, α ίσ οι α. 2. Φέ ο ΑΠ1// ο έ ι ο ί ο Α Π1. Α ό ο 1, 4 σ ο Π1 αι ο ί ο έσο ς Α , φέ ο σ ΑΠ1 αι ο έ ς σ α ά ή ς , 1 1 άθ ο έ ι, α ά ο ώ αΑ ά , ΑΠ1 σ ο 1 ο ί αι έσο ς ΑΠ1. 3. ι ή ο Α 4 1 ί αι ο θο ώ ιο α α ό α ο, ο αι 1 άθ 1 1, ιέ αι α ό ο 7 ο ί αι έσο ς 1 4. ο. . 4. Φέ ο Π2//Α , ο έ ι 9 σ ο Π2.

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ο ί ο Π2 , α ό ο έσο ς , φέ ο 3 αι άθ σ α ά ή ςΑ σοσ ίο ί αι έσο ς 2 αι αφού 12 9=// 2, ο 5 ί Πο ίσ α α: 1. οσ ίο ο ής 2 2 7, 3 5, αφού οι 2 ί αι ο ί ο ο ύ ο ο ά ι ο 2. Αφού 1 1// 2 5, 1 3// 2 7 => ο 1 1 3 2 ί 5.

ιβ ιο

σ 3 2 άθ . Αφού α ά ο ώ 5 αι έσο ς 12 9. ο. . 7,

σο άθ 3 5 ί αι α ύ ο 1 4 9 12. αι α α ό α ο.

Π2 ο ί αι αΑά ο 2

ς

Α 4, Α

9,

αφία: 1. Ασ ήσ ις ω ίας σο ϊ ώ ό F.G.M. όσ ις Α. Κα αβία. Α α ίας Αθή αι, . 2. ά ω ία, ό ος Α΄ ι ο ία, ύχος Α΄, Α ισ ί ο Φ. Πά α, Έ οσις έ α, ιβ ιο ίο Α α έ σις Κ σ α . Α. Θ ο α ί , Πα ισ ίο , Αθή αι, . 3. Ασ ήσ ις αι ω ή α α ω ίας, ώ ιος Αθα ασίο σιά ας, .

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Dardanos Theorem revised

23.04.17 Su *

Author: Georgios. Α. Tsiamas, Gte Electrical Mechanical Engineer, National Technical University of Athens – NTUA Gte of C.I.C.A, Sophia Antipolis, Nice, France Scope In a circle ( , R), we consider two chords at random and they are projected to one another. At a second step, the midpoint of each chord is projected to the other chord. It is proved that the midpoint of first chord, if projected to the second chord, falls on the midpoint of the first chord’s projection to the second chord. Similarly, the projection of the second chord’s midpoint to the first chord falls on the midpoint of its projection to the first chord. The Dardanos Theorem The midpoints M1, M3 of any two chords of a circle ( , R) are projected to the midpoints of the chords’ projections to one another.

9

1 5 12

Α

Π1 1

Π2

2 1 2

1

7

3

4

Proof Symbols and terms used: ^B angle B => as a result, // parallel to, =// equal and parallel to. Azan Theorem: In every triangle Α , the midpoint of a side when projected to a second side, its projection falls on the midpoint of the first side’s projection to that second side. 1. Let Α , be two chords of the circle ( , R) and 1 4 be the projection of Α to and 9 12 be the projection of to Α , 1, 3 be the midpoints of Α , , respectively and 7, 5 be the midpoints of 1 4 and 9 12, respectively. 2. We draw ΑΠ1// that cuts 4 at Π1 and thus triangle Α Π1 is defined. We draw 1 1, from 1, midpoint of Α , perpendicular to ΑΠ1 and thus parallel to , which cuts, due to Azan Theorem, ΑΠ1 at 1 that is midpoint of ΑΠ1. 3. Since Α 4 1 is a rectangular parallelogram, the perpendicular 1 1 from 1, passes through point 7 that is the midpoint of 1 4. q.e.d. 4. We draw Π2//Α , that cuts 9 at Π2. 5. In triangle Π2 , we draw 3 2 perpendicular to Π2, from midpoint 3 of , which 3 2 is also perpendicular to its parallel Α , at point 5. Since, due to Azan Theorem 2 is the midpoint of 2 and since 12 9=// 2, => 5 is the midpoint of 12 9. q.e.d.

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Corollaries: 1. The intersection point 2 of 1 7, inscribed quadrilateral 1 4 9 12 since 2. Since 1 1// 2 5, 1 3// 2 7 => 1

5,

3 1 1

is the center of the circumscribed circle of the 3 5 are the perpendicular bisectors of Α 4, Α 9. 2 is a parallelogram.

7, 3

Note: All invoked theorems of the proof come from the mentioned Bibliography below. Bibliography: 1. Ασ ήσ ις ω ίας σο ϊ ώ ό F.G.M. όσ ις Α. Κα αβία. Α α ίας Αθή αι, . 2. ά ω ία, ό ος Α΄ ι ο ία, ύχος Α΄, Α ισ ί ο Φ. Πά α, Έ οσις έ α, ιβ ιο ίο Α α έ σις Κ σ α . Α. Θ ο α ί , Πα ισ ίο , Αθή αι, . 3. Ασ ήσ ις αι ω ή α α ω ίας, ώ ιος Αθα ασίο σιά ας, 2017.

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