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we study elementary geometric operations on triangles defined as follows . let @xmath0 be a triangle , and @xmath1 be a real number . let @xmath2 , and @xmath3 be division points of the edges @xmath4 , and @xmath5 by @xmath6 respectively , namely , @xmath7 let @xmath8 ( @xmath9 or @xmath10 ) be the intersection of the lines @xmath11 and @xmath12 ( @xmath12 and @xmath13 or @xmath13 and @xmath11 respectively ) . define _ equisection operators _ @xmath14 and @xmath15 , where @xmath15 can be defined when @xmath16 , by @xmath17 the operators @xmath14 have been studied in articles such as @xcite , _ et . al . _    ]    0.4 cm    ]    in this note we study the equivalence relation ( denoted by @xmath18 ) of the set of triangles ( denoted by @xmath19 ) generated by similarity and @xmath20 , which we shall call _ equisectional equivalence_. the equivalence relation generated by similarity and @xmath21 shall be called _ rational equisectional equivalence _ and denoted by @xmath22 . we say two triangles @xmath23 and @xmath24 are _ equisectionally equivalent _ ( or _ rational equisectionally equivalent _ ) if @xmath25 ( or @xmath26 respectively ) . we remark that we use the term `` similarity '' as the equivalence under orientatipon preserving homothetic transformation in this article . we say two triangles are reversely similar if they are equivalent under orientation reversing homothetic transformation . nakamura and oguiso introduced the moduli space of similarity classes of triangles in @xcite , which is a strong tool for the study of @xmath14 and @xmath15 . using their results ( explained in section [ section_no ] ) , we give ( projective ) geometric characterization of equisectionally equivalent triangles . namely , two triangles with a common base , say @xmath27 , with the third vertices , say @xmath28 and @xmath29 , in the same side of the base are equisectionally equivalent if and only if @xmath28 and @xmath29 are on the same circle of apollonius with foci being two vertices ( denoted by @xmath30 and @xmath31 ) of regular triangles with the common base @xmath27 . therefore , each equisectional equivalence class with a given base @xmath27 corresponds to a circle of apollonius with foci @xmath30 and @xmath31 . it is an element of a hyperbolic pencil of circles defined by @xmath30 and @xmath31 from a projective geometric viewpoint . we then study properties of triangles of the following three special types , right triangles , isosceles triangles , and trianges with sides in arithmetic progression ( which shall be denoted by _ sap _ triangles ) , that appear in the same equisectional equivalence class . there are ( at most ) two similarity classes of such triangles for each type , which are reversely similar in the case of right or sap triangles , or the base angles of which satisfy @xmath32 in the case of isosceles triangles . for each type we explicitly give the ratio @xmath1 such that @xmath14 maps one to the other in the same equisectional equivalence class , which implies that a pair of triangles @xmath23 and @xmath24 of one of the above special types with rational edges satisfies @xmath33 if and only if @xmath34 . we finally study compass and straightedge constructibility of @xmath1 for a given pair of triangles . [ def_alpha ] let @xmath35 be a triangle . let @xmath36 be a half plane containing @xmath28 with boundary the line @xmath37 , and @xmath30 and @xmath38 be two points ( @xmath39 ) such that @xmath40 and @xmath41 are regular triangles . define @xmath42 ( @xmath43 ) and @xmath44 by @xmath45 where @xmath46 means the area of @xmath0 . we remark that both @xmath42 and @xmath44 are independent of the choice of the base of the triangle . a locus of points @xmath47 such that @xmath48 is a given positive constant is a circle , called a _ circle of apollonius with foci @xmath30 and @xmath38_. put @xmath49 note that @xmath50 when @xmath23 is a regular triangle . the quantity @xmath51 takes the value @xmath52 if and only if @xmath23 is a regular triangle , and approaches @xmath53 as @xmath23 becomes thinner and thinner . in that sense , it can be considered as measuring how far a triangle is from a regular triangle . [ main_theorem ] given two triangles @xmath35 and @xmath54 . let @xmath29 be a point in @xmath36 such that @xmath55 is similar to @xmath56 . then the following conditions are equivalent :    1 . @xmath23 is equisectionally equivalent to @xmath57 . @xmath58 , in other words , @xmath29 is on the circle of apollonius with foci @xmath30 and @xmath59 that passes through @xmath28 . 3 .   @xmath60 . 4 .   let @xmath61 and @xmath62 be points in @xmath36 such that @xmath63 and @xmath64 are similar to @xmath0 in such a way that each vertex of @xmath63 or @xmath64 corresponds to a vertex of @xmath0 in the same sequential order through the similarity ( figure [ fig_ap_circ_three_pts ] ) . then @xmath29 is on the circle that passes through @xmath65 , and @xmath62 . when @xmath23 is a regular triangle we agree that the circle through @xmath65 , and @xmath62 consists of a single point .     and @xmath59 ]    0.8 cm     and @xmath59 ]    the set of circles of apollonius with foci @xmath30 and @xmath59 is called a _ hyperbolic pencil _ of circles defined by @xmath30 and @xmath38 ( or a _ poncelet pencil _ with _ limit points _ ( or _ poncelet points _ ) @xmath30 and @xmath38 ) . it consists of circles that are orthogonal to any circle passing through @xmath30 and @xmath38 ( figure [ pencil ] left ) . a set of circles through @xmath30 and @xmath38 is called an _ elliptic pencil _ ( or a _ pencil of circles with base points _ ) . let @xmath67 be the set of similarity classes of triangles and @xmath68 $ ] denote the similarity class of a triangle @xmath23 . nakamura and oguiso s result implies that the sets of similarity classes of equisectionally equivalent triangles form a codimension @xmath53 foliation of @xmath67 with a unique singularity @xmath52 that corresponds to regular triangles . we study the intersection of each leaf and another codimension @xmath53 subspace of @xmath67 which is the set of similarity classes of one of the following three special triangles , isosceles triangles , right triangles , and sap triangles ( i.e. , triangles with sides in arithmetic progression ) ( the reader is referred to @xcite for the properties of sap triangles ) . [ cor_right ] two right triangles are equisectionally equivalent if and only if they are either similar or reversely similar . any triangle @xmath23 is equisectionally equivalent to a right triangle if and only if @xmath69 .    [ cor_sap ] two sap triangles are equisectionally equivalent if and only if they are either similar or reversely similar . any trianle is equisectionally equivalent to such a triangle . [ prop_isosceles ] two isosceles triangles @xmath23 and @xmath24 are equisectionally equivalent if and only if either they are similar or the base angles satisfy @xmath70 . when @xmath68\ne[\delta']$ ] , @xmath71=[\delta']$ ] if and only if @xmath72 or @xmath73 . any triangle is equisectionally equivalent to an isosceles triangle . corollaries [ cor_right ] and [ cor_sap ] imply the importance of an equisectional operator that maps a similarity class of a given triangle to that of its mirror image . [ lemma ] let @xmath74 denote a triangle with side lengths @xmath75 in the anti - clockwise order . then @xmath76=[\delta_{ba}]$ ] if and only if @xmath77    [ thm_rationality ] suppose @xmath23 and @xmath24 are isosceles ( or right or sap ) triangles such that all the sides are rational numbers . if @xmath78 then @xmath1 is also rational , namely , @xmath33 if and only if @xmath34 . nakamura and oguiso gave a bijection between @xmath67 and the open unit disc @xmath79 in @xmath80 , and showed that the set of equisection operators @xmath81 acts on @xmath79 as rotations . let us first introduce the result of nakamura - oguiso @xcite . we work in the complex plane @xmath80 . let @xmath82 be the upper half plane @xmath83 , @xmath84 and @xmath79 be the open unit discs in variables @xmath85 and @xmath86 respectively . put @xmath87 . define @xmath88 , and @xmath89 by @xmath90 then we have the following commutative diagram :    @xmath91    let us fix the base of a triangle to be @xmath92 $ ] so that a triangle can be identified by the vertex @xmath93 . suppose @xmath35 is similar to the triangle @xmath94 . since the three choices of the base , @xmath95 , or @xmath5 corresponds to @xmath96 , or @xmath97 , there is a bijection @xmath98 ( @xcite ) given by @xmath99)=g\circ f(z)=\left(\frac{z-\rho}{z-\rho^{-1}}\right)^3\,.\ ] ] let us call @xmath79 the _ nakamura - oguiso moduli space _ of the similarity classes of triangles .    from the construction of the moduli space and the property of linear fractional transformations , it follows that the set of isosceles triangles is expressed by a real axis in @xmath79 ( explained in section [ section_proofs ] ) , and reversely similar triangles by complex conjugate numbers , as was pointed out in @xcite . the equisection operators and the equisectional equivalence relation on @xmath67 , denoted by the same symbols , can be naturally induced from those on @xmath19 . we shall express the operators on @xmath79 given by @xmath100 and @xmath101 simply by @xmath14 and @xmath15 respectively . since our @xmath14 and @xmath15 are equal to @xmath102 and @xmath103 in @xcite respectively , theorem 1 of @xcite implies    [ thm_no](@xcite ) the operator @xmath14 acts on @xmath79 as a rotation by angle @xmath104 and @xmath15 by angle @xmath105 . theorem [ thm_no ] implies that @xmath18 is in fact an equivalence relation , that @xmath33 if and only if there is a real numbers @xmath1 such that @xmath24 is similar to @xmath106 , and that the equivalence relation generated by similarity and @xmath107 is identical with the equisectional equivalence . the following corollaries can be obtained by simple computation . [ cor_t_q_id](@xcite )    1 . @xmath108 if and only if @xmath109 , or @xmath53 . 2 .   @xmath110 and @xmath111 if and only if @xmath72 or @xmath73 . [ cor_t_q_equal ] @xmath112 if and only if @xmath113    [ cor_t_q_inverse](@xcite ) @xmath114 if and only if @xmath115    the six functions of @xmath1 which appear in the right hand sides in corollaries [ cor_t_q_equal ] and [ cor_t_q_inverse ] form a non - abelian group with the operation of composition , which is isomorphic to the full permutation group of three elements .    [ cor_t_q_composition ] given real numbers @xmath1 and @xmath116 . @xmath117 if and only if @xmath118    the formula implies that @xmath119 is the rotation by angle @xmath120.\ ] ] since @xmath121 \hspace{0.7cm}(\mbox{modulo } \>2\pi),\ ] ] substitution @xmath122 and @xmath123 implies that if we put @xmath124 then @xmath117 . the other two values for @xmath125 can be obtained by corollary [ cor_t_q_equal ] . corollaries [ cor_t_q_id ] ( 1 ) , [ cor_t_q_inverse ] , and [ cor_t_q_composition ] show that the rational equisectional equivalence @xmath22 is in fact an equivalence relation . in what follows , we restrict ourselves to the case of non - regular triangles . * proof of theorem * [ main_theorem ] . * *   theorem [ thm_no ] shows that a set of equisectionally equivalent triangles corresponds to a circle with center @xmath52 in @xmath79 . therefore , the formula implies @xmath126\sim[\triangle z'01]\longleftrightarrow \left|\left(\frac{z-\rho}{z-\rho^{-1}}\right)^3\right|=\left|\left(\frac{z'-\rho}{z'-\rho^{-1}}\right)^3\right| \longleftrightarrow \frac{|z-\rho|}{|z-\rho^{-1}|}=\frac{|z'-\rho|}{|z'-\rho^{-1}|},\ ] ] which proves the equivalence between ( 1 ) and ( 2 ) .    since @xmath0 , @xmath127 , and @xmath128 are similar , the above argument implies that a circle of apollonius with foci @xmath129 and @xmath130 that passes through @xmath28 also passes through @xmath61 and @xmath62 , which proves the equivalence of ( 1 ) and ( 4 ) .    the equivalence between ( 2 ) and ( 3 ) follows from computation . since @xmath131 @xmath132 which , translated to a scale - invariant statement , is equivalent to ( 3 ) . we remark that the formula implies @xmath133 . let us give projective geometric explanation of the equivalence between ( 1 ) and ( 2 ) . the set of lines through @xmath52 , which are considered as circles through @xmath52 and @xmath134 , is an elliptic pencil of circles defined by @xmath52 and @xmath134 , and the set of concentric circles with center @xmath52 is a hyperbolic pencil of circles defined by @xmath52 and @xmath134 . they are mutually orthogonal . a linear fractional transformation is a conformal map ( i.e. , it preserves the angles ) that maps circles ( which include lines that can be considered as circles through @xmath134 ) to circles , and hence it maps an elliptic pencil ( or a hyperbolic pencil ) of circles defined by a pair of points to an elliptic pencil ( or a hyperbolic pencil ) defined by a pair of corresponding points .    since @xmath135 is a linear fractional transformation which maps @xmath52 and @xmath134 to @xmath129 and @xmath130 , it maps the set of lines through @xmath52 to an elliptic pencil consisting of the circles through @xmath129 and @xmath130 , and the set of concentric circles with center @xmath52 to a hyperbolic pencil defined by @xmath129 and @xmath130 , which consists of circles of apollonius with foci @xmath129 and @xmath130 ( figure [ pencil ] ) . now it follows from the construction of nakamura - oguiso moduli space that the vertices @xmath136 of equisectionally equivalent triangles @xmath94 form a circle of apollonius with foci @xmath129 and @xmath130 , which proves the equivalence between ( 1 ) and ( 2 ) .     and @xmath130 , and a pair of a hyperbolic pencil ( right , black ) and an elliptic pencil ( right , blue ) of circles defined by @xmath52 and @xmath134 . the former is mapped to the latter by a linear fractional transformation that maps @xmath129 and @xmath130 to @xmath52 and @xmath134 respectively . in each case , a circle in a hyperbolic pencil is orthogonal to a circle in an elliptic pencil . ]    in what follows , it sometimes makes things easier to work in a a fundamental domain of @xmath137 , @xmath138 which corresponds to studying triangles with the longest edge ( one of the longest edges ) being fixed to @xmath139 $ ] . let us explain why the isosceles triangles corresponds to @xmath140 in the real axis in the nakamura - oguiso moduli space @xmath79 . [ def_gamma ] let @xmath141 be a circle through three points @xmath142 , and @xmath143 . when one of the three points is @xmath134 , @xmath141 is a line . we assume that @xmath141 is oriented by the cyclic order of @xmath142 , and @xmath143 .    in @xmath144 , a vertex of an isosceles triangle lies either on the line @xmath145 or on the circle @xmath146 . since @xmath147 is a linear fractional transformation with @xmath148 it maps a circle @xmath146 ( @xmath149 ) to the real axis ( @xmath150 ) , another circle @xmath151 ( @xmath152 ) to a line joining @xmath52 and @xmath153 ( @xmath154 ) , a line @xmath145 ( @xmath155 ) to a line segment joining @xmath52 and @xmath129 ( @xmath156 ) , the real axis ( @xmath150 ) to the unit circle ( @xmath157 ) , and @xmath144 to one third of the open unit disc @xmath158 . therefore , the images of @xmath159 of @xmath160 and @xmath161 are @xmath162 and @xmath163 $ ] respectively . * proof of proposition * [ prop_isosceles ] . * *   we work in the nakamura - oguis moduli space @xmath79 . the isosceles triangles correspond to the real axis in @xmath79 . each circle with center @xmath52 , which corresponds to a set of equisectionally equivalent non - regular triangles , intersects the real axis in two points . it proves the third statement . it also proves that if @xmath14 satisfies @xmath78 with two isosceles triangles @xmath23 and @xmath24 then either @xmath68=[\delta']$ ] or @xmath14 is a rotation by @xmath164 on @xmath79 . theorem [ thm_no ] shows that @xmath14 is a rotation by angle @xmath165 for some @xmath166 if and only if @xmath72 or @xmath73 , which proves the second statement . the equation @xmath70 follows directly from the fact that @xmath167=t_{1/3}([\delta])$ ] as is illustrated in figure [ isosceles3 ] .    ] * proof of corollary * [ cor_right ] . * *   in the fundamental domain @xmath144 , a vertex @xmath136 of a right triangle @xmath94 lies on the upper half hemi - circle @xmath168 , which intersects any circle of apollonius in at most two points , which are symmetric in the line @xmath169 . the extremal value of @xmath51 is given by a right isosceles triangle . * proof of corollary * [ cor_sap ] . * *   in the fundamental domain @xmath144 , a non - regular triangle such that the ratio of the edge lengths is @xmath170 corresponds to @xmath171 let @xmath172 be a curve @xmath173 . we show that each of @xmath172 , say @xmath174 , intersects any circle of apollonius with foci @xmath129 and @xmath130 , @xmath175 , in exactly one point . firstly , since @xmath174 is an open curve joining @xmath129 and a point @xmath176 on the real axis , it must intersect @xmath175 . secondly , if we put @xmath177 , ( or @xmath178 ) of @xmath159 is the intersection of @xmath79 and the upper half plane ( or the lower half plane respectively ) . ] then @xmath179 , and hence @xmath180 . let @xmath181 , then , as @xmath182 is an increasing function and @xmath183 a decreasing function of @xmath184 , we have @xmath185 on @xmath174 , whereas @xmath186 can be expressed as a graph of an increasing function . therefore @xmath174 intersects @xmath175 in at most one point . we remark that the statement can also be proved by computation showing that @xmath187 is a monotonely increasing function of @xmath184 with @xmath188 and @xmath189 . * proof of lemma * [ lemma ] . * *   suppose @xmath190 is expressed by a complex number @xmath191 @xmath192 in the fundamental domain @xmath144 . since @xmath193 and @xmath194 , @xmath195 and @xmath196 satisfy @xmath197 y^2&=&\displaystyle \frac14\left(b^2-(a-1)^2\right)\left((a+1)^2-b^2\right ) . \end{array } \label{xy_ab}\ ] ] let @xmath198 and @xmath199 . then @xmath200 substitution of gives @xmath201 since @xmath74 is a mirror image of @xmath190 , we have @xmath202 , and since @xmath203 , we have @xmath204    on the other hand , @xmath14 acts on @xmath79 as a rotation by @xmath205 , where @xmath206 . suppose @xmath207 it means @xmath208 , which implies @xmath209 modulo @xmath164 and hence @xmath210 modulo @xmath211 , which implies @xmath212 , i.e. , @xmath213)=[\delta_{ba}]$ ] . the equation gives @xmath214 the other two values follow from the above by corollary [ cor_t_q_equal ] . * proof of theorem * [ thm_rationality ] . * *   when @xmath68=[\delta']$ ] the conclusion follows from corollary [ cor_t_q_id ] . when @xmath68\ne[\delta']$ ] the conclusion follows from proposition [ prop_isosceles ] for isosceles triangles and from corollaries [ cor_right ] , [ cor_sap ] and lemma [ lemma ] for right and sap triangles . [ constructibility ] given two triangles @xmath35 and @xmath215 . whether @xmath23 is equisectionally equivalent to @xmath24 or not can be determined using a straightedge and compass , and if the answer is affirmative , a real number @xmath1 such that @xmath216)=[\delta']$ ] is compass - and - straightedge constructible . 1 .   two points @xmath30 and @xmath59 ( @xmath39 ) such that both @xmath40 and @xmath217 are regular triangles . 2 .   a vertex @xmath29 in @xmath36 such that @xmath218=[\triangle a'b'c']$ ] . 3 .   an oriented circle @xmath219 and another oriented circle @xmath220 ( see definition [ def_gamma ] ) . 4 .   a circle of apollonius @xmath175 with foci @xmath30 and @xmath59 that passes through @xmath28 , since the center is the intersection of a bisector of the edge @xmath27 and a tangent line of @xmath221 at point @xmath28 . a decision whether @xmath33 or not , since the answer is affirmative if and only if @xmath222 .      1 .   the signed angle @xmath223 ( @xmath224 ) at point @xmath30 from the oriented circle @xmath219 to @xmath220 at least one @xmath166 such that @xmath225 3 . the value @xmath1 which is given by @xmath226,\ ] ] where @xmath227 is given by the proceeding step . we explain the process ( 6),(7),(8 ) . by working in the fundamental domain @xmath144 , we may assume that @xmath228 , and @xmath229 . by formulae and , we want a real number @xmath1 such that @xmath230 since the right hand side divided by @xmath231 is equal to the singed angle @xmath232 at @xmath52 from the oriented line @xmath233 to the oriented line @xmath234 , and a linear fractional transformation @xmath135 maps @xmath235 , and @xmath134 to @xmath236 , and @xmath130 respectively , the signed angle @xmath232 is equal to the signed angle @xmath223 at point @xmath129 from the oriented circle @xmath237 to the oriented circle @xmath238 . if @xmath1 is given by , then @xmath239 and therefore , @xmath240 ( modulo @xmath211 ) , which means .
we study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio . using the moduli space of similarity classes of triangles introduced by nakamura and oguiso , 
 we give characterization of equivalent triangles in terms of circles of apollonius ( or hyperbolic pencil of circles ) and properties of special equivalent triangles . 
 we also study rationality problem and constructibility problem . 
 2010 _ mathematics subject classification _ : 51m04 .