AN APPLICATION OF THE GENERALIZATION OF CEVA’S THEOREM

AN APPLICATION OF THE GENERALIZATION OF CEVA’S THEOREM Florentin Smarandache University of New Mexico 200 College Road Gallup, NM 87301, USA E-mail: smarand@unm.edu Theorem: Let us consider a polygon A1 A2 ...An inserted in a circle. Let s and t be two non zero natural numbers such that 2s + t = n . By each vertex Ai passes a line di which intersects the lines Ai + s Ai + s +1 ,...., Ai + s + t −1 Ai + s + t at the points M i,i + s ,..., M i + s + t −1 respectively and the circle at the point M i' . Then one has: n i + s + t −1

∏∏ i =1

j =i + s

M ij Aj M ij Aj +1

n

M i' Ai + s

i =1

M i' Ai + s +t

=∏

.

Proof: Let i be fixed.

1) The case where the point M i,i + s is inside the circle.

There are the triangles

Ai M i,i + s Ai + s and

M i' M i,i + s Ai + s +1 similar, since the angles

M i,i + s Ai Ai + s and M i,i + s Ai + s +1 M i' on one side, and Ai M i,i + s Ai + s and Ai + s +1 M i,i + s M i' are equal. It results from it that: M i,i + s Ai AA (1) = 'i i + s M i,i + s Ai + s +1 M i Ai + s +1

Ai+s+1

Ai MMM Mi,i+s

M i' Ai+s

1

In a similar manner, one shows that the triangles M i,i + s Ai Ai + s +1 and M i,i + s Ai + s M i' are similar, from which: M i,i + s Ai AA = i ' i + s +1 . Dividing (1) by (2) we obtain: (2) M i,i + s Ai + s M i Ai + s (3)

M i,i + s Ai + s

=

M i,i + s Ai + s +1

M i' Ai + s ' i

M Ai + s +1

⋅

Ai Ai + s .

Ai Ai + s +1

2) The case where M i,i + s is exterior to the circle is similar to the first, because the triangles (notations as in 1) are similar also in this new case. There are the same interpretations and the same ratios; therefore one has also the relation (3).

Ai+s+1

Ai+s

Ai

M i'

Mi,i+s

Let us calculate the product: i + s + t −1

M ij A j

j =i+ s

M ij A j +1

∏

=

M i' Ai + s

⋅

=

i + s + t −1

∏

j =i+ s

M i' Ai + s +1

M i' Ai + s +1 M i' Ai + s + 2

⎛ M i' A j Ai A j ⎞ ⋅ ⎜ ' ⎟= ⎜⎝ M A ⎟ A A i j +1 ⎠ i j +1 ⋅⋅⋅

M i' Ai + s + t −1 M i' Ai + s + t

⋅

Ai Ai + s Ai Ai + s +1 Ai Ai + s + t −1 M i' Ai + s Ai Ai + s ⋅ ⋅ ⋅⋅⋅ = ' ⋅ Ai Ai + s +1 Ai Ai + s + 2 Ai Ai + s + t M i Ai + s + t Ai Ai + s + t Therefore the initial product is equal to:

2

n ⎛ M i' Ai + s Ai Ai + s ⎞ M i' Ai + s ⋅ ∏ ⎜ ' ⎟ =∏ Ai Ai + s + t ⎠ i =1 M i' Ai + s + t i = 1 ⎝ M i Ai + s + t since: n

n

Ai Ai + s

∏AA i =1

⋅

i

i + s+t

=

A1 A1+ s A2 A2 + s AA ⋅ ⋅ ⋅ ⋅ s 2s ⋅ A1 A1+ s + t A2 A2 + s + t As +1 A1

As + 2 A2 s + 2 A A A A AA A A ⋅ ⋅ ⋅ s + t n ⋅ s + t +1 1 ⋅ s + t + 2 2 ⋅ ⋅ ⋅ n s = 1 As + 2 A2 As + t At As + t +1 At +1 As + t + 2 At + 2 An As + t

(by taking into account the fact that 2s + t = n ).

Consequence 1: If there is a polygon A1 A2 ,...., A2 s −1 inscribed in a circle, and from each vertex Ai one traces a line di which intersects the opposite side Ai + s −1 Ai + s in M i and the circle in M i' then: n M i' Ai + s −1 M i Ai + s −1 = ∏ ∏ ' i = 1 M i Ai + s i = 1 M i Ai + s n

n +1 .

2 If one makes s = 1 in this consequence, one finds the mathematical note from [1], pages 35-37.

Application: If in the theorem, the lines di are concurrent, one obtains: In fact for t = 1 , one has n odd and s =

n

M i' Ai + s

∏M A i =1

' i

= (−1)n (For this, see [2]).

i + s+t

Bibliography: [1] Dan Barbilian, Ion Barbu – “Pagini inedite”, Editura Albatros, Bucharest, 1981 (Ediţie îngrijită de Gerda Barbilian, V. Protopopescu, Viorel Gh. Vodă).

[2] Florentin Smarandache – “Généralisation du théorème de Ceva”.

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