diff options
Diffstat (limited to 'Master/texmf-dist/doc/latex/tkz-elements/TKZdoc-elements-classes-circle.tex')
-rw-r--r-- | Master/texmf-dist/doc/latex/tkz-elements/TKZdoc-elements-classes-circle.tex | 375 |
1 files changed, 218 insertions, 157 deletions
diff --git a/Master/texmf-dist/doc/latex/tkz-elements/TKZdoc-elements-classes-circle.tex b/Master/texmf-dist/doc/latex/tkz-elements/TKZdoc-elements-classes-circle.tex index 85e73595189..2d5665f879b 100644 --- a/Master/texmf-dist/doc/latex/tkz-elements/TKZdoc-elements-classes-circle.tex +++ b/Master/texmf-dist/doc/latex/tkz-elements/TKZdoc-elements-classes-circle.tex @@ -38,7 +38,8 @@ Three attributes are used (south, west, radius). \begin{minipage}{0.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .5 z.a = point: new (1, 1) z.b = point: new (5, 4) @@ -48,7 +49,7 @@ Three attributes are used (south, west, radius). r = C.ab.radius z.c = C.ab.opp z.r,z.t = get_points (C.ab.ct : ortho_from (z.b)) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(a,b,c,s,w) @@ -60,7 +61,8 @@ Three attributes are used (south, west, radius). \end{Verbatim} \end{minipage} \begin{minipage}{0.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () scale = .5 z.a = point: new (1, 1) z.b = point: new (5, 4) @@ -70,16 +72,16 @@ Three attributes are used (south, west, radius). r = C.ab.radius z.c = C.ab.opp z.r,z.t = get_points (C.ab.ct : ortho_from (z.b)) -\end{tkzelements} +} -\emph{\begin{tikzpicture} +\hfill\begin{tikzpicture} \tkzGetNodes \tkzDrawPoints(a,b,c,s,w) \tkzLabelPoints(a,b,c,s,w) \tkzDrawCircle(a,b) \tkzDrawSegments(a,b r,t b,c) \tkzLabelSegment[sloped](a,b){ab = \tkzUseLua{r}} -\end{tikzpicture}} +\end{tikzpicture} \end{minipage} % subsubsection example_circle_attributes (end) @@ -127,7 +129,7 @@ Three attributes are used (south, west, radius). \midrule \textbf{Miscellaneous} &&\\ \midrule -\Imeth{circle}{power (pt)} &| r = C.OA: power (z.M)| & [\ref{par:power_v1} ; \ref{par:power_v2} ; \ref{sub:apollonius_circle_v1_with_inversion} ] \\ +\Imeth{circle}{power (pt)} &| r = C.OA: power (z.M)| & [\ref{ssub:power_v1} ; \ref{ssub:power_v2} ; \ref{sub:apollonius_circle_v1_with_inversion} ] \\ \Imeth{circle}{in\_out (pt)} & |C.OA : in_out (z.M)| & [\ref{ssub:in_out_for_circle_and_disk}] \\ \Imeth{circle}{in\_out\_disk (pt)} & |C.OA : in_out_disk (z.M)| & [\ref{ssub:in_out_for_circle_and_disk}] \\ \Imeth{circle}{draw ()} & for further use &\\ @@ -145,11 +147,12 @@ A circle is defined by its centre and a point through which it passes. \vspace{6pt} \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point: new (0,0) z.A = point: new (2,1) C = circle: new (z.O , z.A) -\end{tkzelements} +} \begin{tikzpicture}[gridded] \tkzGetNodes \tkzDrawCircles(O,A) @@ -159,11 +162,12 @@ C = circle: new (z.O , z.A) \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point: new (0,0) z.A = point: new (2,1) C = circle: new (z.O , z.A) -\end{tkzelements} +} \begin{center} \begin{tikzpicture}[gridded] \tkzGetNodes @@ -186,12 +190,13 @@ We define a circle with its centre and radius. \vspace{6pt} \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point: new (0,0) z.A = point: new (2,1) C = circle: radius (z.A , math.sqrt(5)) z.T = C.through -\end{tkzelements} +} \begin{tikzpicture}[gridded] \tkzGetNodes \tkzDrawCircles(A,T) @@ -201,12 +206,13 @@ z.T = C.through \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} - \begin{tkzelements} + \directlua{% +init_elements () z.O = point: new (0,0) z.A = point: new (2,1) C = circle: radius (z.A , math.sqrt(5)) z.T = C.through - \end{tkzelements} + } \begin{center} \begin{tikzpicture}[gridded] \tkzGetNodes @@ -226,13 +232,14 @@ A circle is defined by two points at the ends of one of its diameters. \vspace{6pt} \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point: new (0,0) z.B = point: new (2,1) C = circle: diameter (z.A , z.B) z.O = C.center z.T = C.through -\end{tkzelements} +} \begin{tikzpicture}[gridded] \tkzGetNodes \tkzDrawCircles(O,T) @@ -242,13 +249,14 @@ z.T = C.through \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point: new (0,0) z.B = point: new (2,1) C = circle: diameter (z.A , z.B) z.O = C.center z.T = C.through -\end{tkzelements} +} \begin{center} \begin{tikzpicture}[gridded] \tkzGetNodes @@ -267,12 +275,13 @@ This method is used to define a point that is diametrically opposed to a point o \vspace{6pt} \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point: new (0,0) z.O = point: new (2,1) C = circle: new (z.O , z.A) z.B = C : antipode (z.A) -\end{tkzelements} +} \begin{tikzpicture}[gridded] \tkzGetNodes \tkzDrawCircles(O,A) @@ -282,12 +291,13 @@ z.B = C : antipode (z.A) \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point: new (0,0) z.O = point: new (2,1) C = circle: new (z.O , z.A) z.B = C : antipode (z.A) -\end{tkzelements} +} \begin{center} \begin{tikzpicture}[gridded] \tkzGetNodes @@ -311,13 +321,14 @@ The definition I use here is more general: the defined point is simply the point \vspace{6pt} \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point: new (0,0) z.O = point: new (2,1) C = circle: new (z.O , z.A) z.B = C : point (0.25) z.M = C : midarc (z.A,z.B) -\end{tkzelements} +} \begin{tikzpicture}[gridded] \tkzGetNodes \tkzDrawCircles(O,A) @@ -327,13 +338,14 @@ z.M = C : midarc (z.A,z.B) \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point: new (0,0) z.O = point: new (2,1) C = circle: new (z.O , z.A) z.B = C : point (0.25) z.M = C : midarc (z.A,z.B) -\end{tkzelements} +} \begin{center} \begin{tikzpicture}[gridded] \tkzGetNodes @@ -358,14 +370,15 @@ If $r=.5$ the defined point is diametrically opposed to $A$, the angle $\widehat \vspace{6pt} \begin{minipage}{.5\textwidth} \begin{Verbatim} - \begin{tkzelements} + \directlua{% +init_elements () z.O = point: new (0,0) z.A = point: new (1,2) C.OA = circle: new (z.O,z.A) z.B = C.OA: point (1/6) z.C = C.OA: point (0.25) z.D = C.OA: point (0.5) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) @@ -375,14 +388,15 @@ If $r=.5$ the defined point is diametrically opposed to $A$, the angle $\widehat \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} - \begin{tkzelements} + \directlua{% +init_elements () z.O = point: new (0,0) z.A = point: new (1,2) C.OA = circle: new (z.O,z.A) z.B = C.OA: point (1/6) z.C = C.OA: point (0.25) z.D = C.OA: point (0.5) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} \tkzGetNodes @@ -409,14 +423,15 @@ The \code{inversion} method can be used on a point, a group of points, a line or \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.o = point: new (-1,2) z.a = point: new (2,1) C.oa = circle: new (z.o,z.a) z.c = point: new (3,4) z.d = C.oa: inversion (z.c) p = C.oa: power (z.c) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(o,a) @@ -429,7 +444,8 @@ The \code{inversion} method can be used on a point, a group of points, a line or \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () scale =.75 z.o = point: new (-1,2) z.a = point: new (2,1) @@ -437,7 +453,7 @@ The \code{inversion} method can be used on a point, a group of points, a line or z.c = point: new (3,4) z.d = C.oa: inversion (z.c) p = C.oa: power (z.c) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -460,7 +476,8 @@ The result is either a straight line or a circle. \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.o = point: new (-1,1) z.a = point: new (1,3) C.oa = circle: new (z.o,z.a) @@ -469,7 +486,7 @@ The result is either a straight line or a circle. L.cd = line: new (z.c,z.d) C.OH = C.oa: inversion (L.cd) z.O,z.H = get_points(C.OH) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(o,a O,H) @@ -480,7 +497,8 @@ The result is either a straight line or a circle. \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () z.o = point: new (-1,1) z.a = point: new (1,3) C.oa = circle: new (z.o,z.a) @@ -489,7 +507,7 @@ The result is either a straight line or a circle. L.cd = line: new (z.c,z.d) C.OH = C.oa: inversion (L.cd) z.O,z.H = get_points(C.OH) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -511,7 +529,8 @@ The result is either a straight line or a circle. \begin{minipage}{.55\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .7 z.o,z.a = point: new (-1,3),point: new (2,3) z.c = point: new (-2,1) @@ -528,7 +547,7 @@ if obj.type == "line" then z.p,z.q = get_points(obj) else z.f,z.b = get_points(obj) end color = "orange" -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[black](o,a) @@ -541,7 +560,8 @@ color = "orange" \end{Verbatim} \end{minipage} \begin{minipage}{.45\textwidth} - \begin{tkzelements} + \directlua{% +init_elements () scale = .7 z.o,z.a = point: new (-1,3),point: new (2,3) z.c = point: new (-2,1) @@ -558,7 +578,7 @@ color = "orange" then z.p,z.q = get_points(obj) else z.f,z.b = get_points(obj) end color = "orange" - \end{tkzelements} + } \begin{center} \begin{tikzpicture} @@ -582,8 +602,9 @@ Circles are geometrically similar to one another and mirror symmetric. Hence, a \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} - scale = 0.75 +\directlua{% +init_elements () + scale = 0.7 z.A = point : new ( 0 , 0 ) z.a = point : new ( 2 , 2 ) z.B = point : new ( 5 , 2 ) @@ -594,7 +615,7 @@ z.I = C.Aa : internal_similitude (C.Bb) L.TA1,L.TA2 = C.Aa : tangent_from (z.I) z.A1 = L.TA1.pb z.A2 = L.TA2.pb -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,a B,b) @@ -604,8 +625,9 @@ z.A2 = L.TA2.pb \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} - scale = .75 +\directlua{% +init_elements () + scale = .7 z.A = point : new ( 0 , 0 ) z.a = point : new ( 2 , 2 ) z.B = point : new ( 5 , 2 ) @@ -616,7 +638,7 @@ z.I = C.Aa : internal_similitude (C.Bb) L.TA1,L.TA2 = C.Aa : tangent_from (z.I) z.A1 = L.TA1.pb z.A2 = L.TA2.pb -\end{tkzelements} +} \begin{center} \begin{tikzpicture} \tkzGetNodes @@ -634,46 +656,48 @@ z.A2 = L.TA2.pb \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point : new ( 0 , 0 ) z.a = point : new ( 2 , 2 ) z.B = point : new ( 3 , 2 ) -z.b = point : new ( 4 , 1 ) +z.b = point : new ( 3.5 , 1 ) C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) z.I = C.Aa : external_similitude (C.Bb) L.TA1,L.TA2 = C.Aa : tangent_from (z.I) z.A1 = L.TA1.pb z.A2 = L.TA2.pb -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,a B,b) \tkzDrawPoints(A,a,B,b,I,A1,A2) -\tkzDrawLines[add = .5 and .2](A1,I A2,I) +\tkzDrawLines[add = .25 and .1](A1,I A2,I) \end{tikzpicture} \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () scale = .75 z.A = point : new ( 0 , 0 ) -z.a = point : new ( 2 , 2 ) +z.a = point : new ( 2 , 2 ) z.B = point : new ( 3 , 2 ) -z.b = point : new ( 4 , 1 ) +z.b = point : new ( 3.5, 1 ) C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) z.I = C.Aa : external_similitude (C.Bb) L.TA1,L.TA2 = C.Aa : tangent_from (z.I) z.A1 = L.TA1.pb z.A2 = L.TA2.pb -\end{tkzelements} +} \begin{center} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,a B,b) \tkzDrawPoints(A,a,B,b,I,A1,A2) -\tkzDrawLines[add = .5 and .2](A1,I A2,I) +\tkzDrawLines[add = .25 and .1](A1,I A2,I) \end{tikzpicture} \end{center} \end{minipage} @@ -691,7 +715,9 @@ Here I have also named \code{radical\_center} the point of intersection of the r \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () + scale = .8 z.O = point : new (0,0) z.x = point : new (1,0) z.y = point : new (4,0) @@ -705,12 +731,12 @@ Here I have also named \code{radical\_center} the point of intersection of the r z.bp,z.b = intersection (C.Opy,C.Pz) L.aap = line : new (z.a,z.ap) L.bbp = line : new (z.b,z.bp) - -- z.X = intersection (L.aap,L.bbp) + % z.X = intersection (L.aap,L.bbp) z.X = C.Ox : radical_center(C.Pz,C.Opy) - -- L.OOp = line : new (z.O,z.Op) - -- z.H = L.OOp : projection (z.X) + % L.OOp = line : new (z.O,z.Op) + % z.H = L.OOp : projection (z.X) z.H = C.Ox : radical_center(C.Opy) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,a O',b P,z) @@ -721,7 +747,9 @@ Here I have also named \code{radical\_center} the point of intersection of the r \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% -\begin{tkzelements} +\directlua{% +init_elements () +scale = .8 z.O = point : new (0,0) z.x = point : new (1,0) z.y = point : new (4,0) @@ -738,7 +766,7 @@ L.bbp = line : new (z.b,z.bp) z.X = intersection (L.aap,L.bbp) L.OOp = line : new (z.O,z.Op) z.H = L.OOp : projection (z.X) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -763,7 +791,8 @@ The radical line, also called the radical axis, is the locus of points of equal \label{par:radical_axis_v1} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .75 z.X = point : new (0,0) z.B = point : new (2,2) @@ -782,7 +811,7 @@ L.AB = line : new (z.A,z.B) L.ApBp = line : new (z.Ap,z.Bp) z.M = intersection (L.AB,L.ApBp) z.H = L.XY : projection (z.M) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(X,B Y,A') @@ -793,7 +822,8 @@ z.H = L.XY : projection (z.M) \end{tikzpicture} \end{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .75 z.X = point : new (0,0) z.B = point : new (2,2) @@ -812,7 +842,7 @@ L.AB = line : new (z.A,z.B) L.ApBp = line : new (z.Ap,z.Bp) z.M = intersection (L.AB,L.ApBp) z.H = L.XY : projection (z.M) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -830,7 +860,8 @@ z.H = L.XY : projection (z.M) \label{par:radical_axis_v2} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = 1.25 z.O = point : new (-1,0) z.Op = point : new (4,-1) @@ -847,12 +878,12 @@ _,z.Tp = get_points (L.MTp) L.MK,L.MKp = C.OpD : tangent_from (z.M) _,z.K = get_points (L.MK) _,z.Kp = get_points (L.MKp) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',D) \tkzDrawLine(E,F) - \tkzDrawLine[add=.5 and .5](O,O') + \tkzDrawLine[add=.25 and .25](O,O') \tkzDrawLines[add = 0 and .5](M,T M,T' M,K M,K') \tkzDrawCircle(M,T) \tkzDrawPoints(O,O',T,M,T',K,K') @@ -860,7 +891,8 @@ _,z.Kp = get_points (L.MKp) \end{tikzpicture} \end{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale =1.25 z.O = point : new (-1,0) z.Op = point : new (4,-1) @@ -877,14 +909,14 @@ _,z.Tp = get_points (L.MTp) L.MK,L.MKp = C.OpD : tangent_from (z.M) _,z.K = get_points (L.MK) _,z.Kp = get_points (L.MKp) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',D) \tkzDrawLine(E,F) - \tkzDrawLine[add=.5 and .5](O,O') + \tkzDrawLine[add=.25 and .25](O,O') \tkzDrawLines[add = 0 and .5](M,T M,T' M,K M,K') \tkzDrawCircle(M,T) \tkzDrawPoints(O,O',T,M,T',K,K') @@ -897,7 +929,8 @@ _,z.Kp = get_points (L.MKp) \label{par:radical_axis_v3} \begin{Verbatim} - \begin{tkzelements} + \directlua{% +init_elements () z.O = point : new (0,0) z.B = point : new (4,0) z.Op = point : new (6,0) @@ -910,7 +943,7 @@ _,z.Kp = get_points (L.MKp) _,z.T = get_points (L) L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',B) @@ -923,7 +956,8 @@ _,z.Kp = get_points (L.MKp) \end{tikzpicture} \end{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point : new (0,0) z.B = point : new (4,0) z.Op = point : new (6,0) @@ -936,7 +970,7 @@ L = C.OB : tangent_from (z.M) _,z.T = get_points (L) L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -956,7 +990,8 @@ _,z.Tp = get_points (L) \label{par:radical_axis_v4} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point : new (0,0) z.B = point : new (5,0) z.Op = point : new (3,0) @@ -970,7 +1005,7 @@ _,z.Tp = get_points (L) _,z.T = get_points (L) _,L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(O,B O',B) @@ -983,7 +1018,8 @@ _,z.Tp = get_points (L) \end{tikzpicture} \end{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point : new (0,0) z.B = point : new (5,0) z.Op = point : new (3,0) @@ -997,7 +1033,7 @@ _,z.Tp = get_points (L) _,z.T = get_points (L) _,L = C.OpB : tangent_from (z.M) _,z.Tp = get_points (L) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -1021,7 +1057,8 @@ _,z.Tp = get_points (L) \begin{minipage}{.5\textwidth} \begin{Verbatim} - \begin{tkzelements} + \directlua{% +init_elements () z.A = point: new (0,0) z.B = point: new (1,2) C.AB = circle: new (z.A,z.B) @@ -1031,7 +1068,7 @@ _,z.Tp = get_points (L) L.T1,L.T2 = C.AB : tangent_from (z.C) z.T1 = L.T1.pb z.T2 = L.T2.pb -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(A,B) @@ -1045,7 +1082,8 @@ _,z.Tp = get_points (L) \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} - \begin{tkzelements} + \directlua{% +init_elements () z.A = point: new (0,0) z.B = point: new (1,2) C.AB = circle: new (z.A,z.B) @@ -1055,7 +1093,7 @@ _,z.Tp = get_points (L) L.T1,L.T2 = C.AB : tangent_from (z.C) z.T1 = L.T1.pb z.T2 = L.T2.pb -\end{tkzelements} +} \begin{center} \begin{tikzpicture} \tkzGetNodes @@ -1078,7 +1116,8 @@ _,z.Tp = get_points (L) Let be a tangent common to both circles at $T$ and $T'$ (closest to $C$). Let a secant parallel to this tangent pass through $C$. Then the segment $[TT']$ is seen from the other common point $D$ at an angle equal to half the angle of the two circles. \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point : new ( 0 , 0 ) z.B = point : new ( 5 , 2 ) L.AB = line : new ( z.A , z.B ) @@ -1094,7 +1133,7 @@ Let be a tangent common to both circles at $T$ and $T'$ (closest to $C$). Let a _,z.M = intersection (L.mm, C.AC) z.Mp = intersection (L.mm, C.BC) _,z.D = intersection (C.AC,C.BC) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(A,C B,C) @@ -1109,7 +1148,8 @@ Let be a tangent common to both circles at $T$ and $T'$ (closest to $C$). Let a \end{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point : new ( 0 , 0 ) z.B = point : new ( 5 , 2 ) L.AB = line : new ( z.A , z.B ) @@ -1125,7 +1165,7 @@ L.mm = L.TTp : ll_from (z.C) _,z.M = intersection (L.mm, C.AC) z.Mp = intersection (L.mm, C.BC) _,z.D = intersection (C.AC,C.BC) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -1150,14 +1190,15 @@ In geometry, two circles are said to be orthogonal if their respective tangent l This method determines a circle with a given centre, orthogonal to a circle that is also given. \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .6 z.C_1 = point: new (0,0) z.C_2 = point: new (8,0) z.A = point: new (5,0) C = circle: new (z.C_1,z.A) z.S,z.T = get_points (C: orthogonal_from (z.C_2)) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles(C_1,T C_2,T) @@ -1176,14 +1217,15 @@ This method determines a circle with a given centre, orthogonal to a circle that \end{tikzpicture} \end{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .6 z.C_1 = point: new (0,0) z.C_2 = point: new (8,0) z.A = point: new (5,0) C = circle: new (z.C_1,z.A) z.S,z.T = get_points (C: orthogonal_from (z.C_2)) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -1210,7 +1252,8 @@ z.S,z.T = get_points (C: orthogonal_from (z.C_2)) \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point: new (0,1) z.A = point: new (1,0) z.z1 = point: new (-1.5,-1.5) @@ -1218,7 +1261,7 @@ z.S,z.T = get_points (C: orthogonal_from (z.C_2)) C.OA = circle: new (z.O,z.A) C = C.OA: orthogonal_through (z.z1,z.z2) z.c = C.center -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) @@ -1228,7 +1271,8 @@ z.S,z.T = get_points (C: orthogonal_from (z.C_2)) \end{tikzpicture} \end{Verbatim} \end{minipage} -\begin{minipage}{.5\textwidth}\begin{tkzelements} +\begin{minipage}{.5\textwidth}\directlua{% +init_elements () z.O = point: new (0,1) z.A = point: new (1,0) z.z1 = point: new (-1.5,-1.5) @@ -1236,7 +1280,7 @@ z.S,z.T = get_points (C: orthogonal_from (z.C_2)) C.OA = circle: new (z.O,z.A) C = C.OA: orthogonal_through (z.z1,z.z2) z.c = C.center -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -1274,7 +1318,8 @@ We can obtain the centers of similarity of these two circles by constructing $EH \begin{minipage}{.4\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .8 z.A = point : new ( 1 , 0 ) z.B = point : new ( 3 , 0 ) @@ -1290,11 +1335,12 @@ C.IT,C.JV = C.AO : midcircle (C.BP) z.I,z.T = get_points (C.IT) z.J,z.V = get_points (C.JV) z.X,z.Y = intersection (C.AO,C.BP) -\end{tkzelements} +} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} - \begin{tkzelements} + \directlua{% +init_elements () scale = .8 z.A = point : new ( 1 , 0 ) z.B = point : new ( 3 , 0 ) @@ -1310,7 +1356,7 @@ z.X,z.Y = intersection (C.AO,C.BP) z.I,z.T = get_points (C.IT) z.J,z.V = get_points (C.JV) z.X,z.Y = intersection (C.AO,C.BP) - \end{tkzelements} + } \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal,thick](A,O B,P) @@ -1337,7 +1383,8 @@ z.X,z.Y = intersection (C.AO,C.BP) \label{midcircle_diameter} \begin{minipage}{.6\textwidth} \begin{Verbatim} - \begin{tkzelements} + \directlua{% +init_elements () scale =.75 z.A = point : new ( 3 , 0 ) z.B = point : new ( 5 , 0 ) @@ -1354,11 +1401,12 @@ z.X,z.Y = intersection (C.AO,C.BP) z.y = C.UR.center C.IT = C.AO : midcircle (C.BP) z.I,z.T = get_points (C.IT) -\end{tkzelements} +} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} - \begin{tkzelements} + \directlua{% +init_elements () scale =.75 z.A = point : new ( 3 , 0 ) z.B = point : new ( 5 , 0 ) @@ -1375,7 +1423,7 @@ z.X,z.Y = intersection (C.AO,C.BP) z.y = C.UR.center C.IT = C.AO : midcircle (C.BP) z.I,z.T = get_points (C.IT) - \end{tkzelements} + } \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal,thick](A,O B,P) @@ -1397,7 +1445,8 @@ $I$ is the center of external similarity of the two given circles. To obtain the \begin{minipage}{.4\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale=.75 local a,b,c,d z.A = point : new ( 0 , 0 ) @@ -1416,11 +1465,12 @@ L.TF = C.Bb : tangent_from (z.I) z.H = intersection (L.TF,C.IT) z.E = intersection (L.TF,C.Aa) z.F=L.TF.pb -\end{tkzelements} +} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () scale=.75 local a,b,c,d z.A = point : new ( 0 , 0 ) @@ -1439,7 +1489,7 @@ L.TF = C.Bb : tangent_from (z.I) z.H = intersection (L.TF,C.IT) z.E = intersection (L.TF,C.Aa) z.F=L.TF.pb -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal,thick](A,a B,b) @@ -1462,7 +1512,8 @@ z.F=L.TF.pb \begin{minipage}{.4\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale=.75 local a,b,c,d z.A = point : new ( 0 , 0 ) @@ -1481,11 +1532,12 @@ L.TF = C.Bb : tangent_from (z.I) z.H = intersection (L.TF,C.IT) z.E = intersection (L.TF,C.Aa) z.F=L.TF.pb -\end{tkzelements} +} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () scale=.75 local a,b,c,d z.A = point : new ( 0 , 0 ) @@ -1504,7 +1556,7 @@ L.TF = C.Bb : tangent_from (z.I) z.H = intersection (L.TF,C.IT) z.E = intersection (L.TF,C.Aa) z.F=L.TF.pb -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircles[teal,thick](A,a B,b) @@ -1524,7 +1576,8 @@ z.F=L.TF.pb \begin{minipage}{.4\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point : new ( 2 , 0 ) z.B = point : new ( 4 , 0 ) z.a = point : new ( 1 , 0) @@ -1533,11 +1586,12 @@ C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) C.IT = C.Aa : midcircle (C.Bb) z.I,z.T = get_points(C.IT) -\end{tkzelements} +} \end{Verbatim} \end{minipage} \begin{minipage}{.6\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point : new ( 2 , 0 ) z.B = point : new ( 4 , 0 ) z.a = point : new ( 1 , 0) @@ -1546,7 +1600,7 @@ C.Aa = circle : new (z.A,z.a) C.Bb = circle : new (z.B,z.b) C.IT = C.Aa : midcircle (C.Bb) z.I,z.T = get_points (C.IT) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes @@ -1569,7 +1623,8 @@ z.I,z.T = get_points (C.IT) \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () scale = .5 z.A = point: new (0,0) z.B = point: new (6,0) @@ -1583,7 +1638,7 @@ z.I,z.T = get_points (C.IT) z.I_c,z.Xc = get_points (C.exc) C.ortho = C.exa : radical_circle (C.exb,C.exc) z.w,z.a = get_points (C.ortho) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawPolygon(A,B,C) @@ -1595,7 +1650,8 @@ z.I,z.T = get_points (C.IT) \end{Verbatim} \end{minipage} -\begin{tkzelements} +\directlua{% +init_elements () scale = .5 z.A = point: new (0,0) z.B = point: new (6,0) @@ -1609,7 +1665,7 @@ z.I,z.T = get_points (C.IT) z.I_c,z.Xc = get_points (C.exc) C.ortho = C.exa : radical_circle (C.exb,C.exc) z.w,z.a = get_points (C.ortho) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -1623,14 +1679,14 @@ z.I,z.T = get_points (C.IT) \end{center} % subsubsection radical_circle (end) -\subsubsection{Method \Imeth{circle}{power(C)}} % (fold) -\label{ssub:method_imeth_circle_power_c} +\subsubsection{Method \Imeth{circle}{power(C)} Power v1} % (fold) +\label{ssub:power_v1} -\paragraph{Power v1} % (fold) -\label{par:power_v1} -\begin{minipage}[t]{.5\textwidth}\vspace{0pt}% + +\begin{minipage}[t]{.45\textwidth}\vspace{0pt}% \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point : new (0,0) z.A = point : new (2,-2) z.M = point : new (-6,0) @@ -1638,7 +1694,7 @@ z.I,z.T = get_points (C.IT) C.OA = circle : new (z.O,z.A) z.Ap = C.OA : antipode (z.A) z.B = intersection (L.AM, C.OA) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) @@ -1650,9 +1706,10 @@ z.I,z.T = get_points (C.IT) \end{tikzpicture} \end{Verbatim} \end{minipage} -\begin{minipage}[t]{.5\textwidth} -\begin{tkzelements} -scale = 1 +\begin{minipage}[t]{.55\textwidth}\vspace{0pt}% +\directlua{% +init_elements () +scale = .75 z.O = point : new (0,0) z.A = point : new (2,-2) z.M = point : new (-6,0) @@ -1660,11 +1717,8 @@ L.AM = line : new (z.A,z.M) C.OA = circle : new (z.O,z.A) z.Ap = C.OA : antipode (z.A) z.B = intersection (L.AM, C.OA) -\end{tkzelements} - - -\begin{center} - \begin{tikzpicture} +} +\begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) \tkzMarkRightAngle[fill=gray!10](A',B,M) @@ -1673,16 +1727,18 @@ z.B = intersection (L.AM, C.OA) \tkzLabelPoints(O,A,A',M,B) \tkzDrawSegments[-Triangle](M,A M,A') \end{tikzpicture} -\end{center} - \end{minipage} -% paragraph power_v1 (end) +% subsubsection power_v1 (end) + +\subsubsection{Method \Imeth{circle}{power(C)} Power v2} % (fold) +\label{ssub:power_v2} +\vspace{6pt} + -\paragraph{Power v2} % (fold) -\label{par:power_v2} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.O = point : new (0,0) z.A = point : new (2,2) z.M = point : new (-1.5,0) @@ -1693,7 +1749,7 @@ z.B = intersection (L.AM, C.OA) z.m = z.M : north(1) L.mM = line : new (z.m,z.M) z.U,z.V = intersection (L.mM,C.OA) -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes \tkzDrawCircle(O,A) @@ -1707,8 +1763,9 @@ z.B = intersection (L.AM, C.OA) \end{Verbatim} \end{minipage} \begin{minipage}[t]{.5\textwidth}\vspace{0pt}% -\begin{tkzelements} -scale = 1 +\directlua{% +init_elements () +scale = .8 z.O = point : new (0,0) z.A = point : new (2,2) z.M = point : new (-1.5,0) @@ -1719,7 +1776,7 @@ _,z.B = intersection (L.AM, C.OA) z.m = z.M : north(1) L.mM = line : new (z.m,z.M) z.U,z.V = intersection (L.mM,C.OA) -\end{tkzelements} +} \begin{center} \begin{tikzpicture} @@ -1735,7 +1792,7 @@ z.U,z.V = intersection (L.mM,C.OA) \end{center} \end{minipage} -% paragraph power_v2 (end) +% subsubsection power_v2 (end) % subsubsection method_imeth_circle_power_c (end) @@ -1744,7 +1801,8 @@ z.U,z.V = intersection (L.mM,C.OA) \begin{minipage}{.5\textwidth} \begin{Verbatim} - \begin{tkzelements} + \directlua{% +init_elements () z.O = point : new (0,0) z.A = point : new (1,2) C.OA = circle : new (z.O,z.A) @@ -1757,10 +1815,11 @@ z.U,z.V = intersection (L.mM,C.OA) BDn = C.OA : in_out_disk (z.N) BCp = C.OA : in_out (z.P) BDp = C.OA : in_out_disk (z.P) - \end{tkzelements} + } \end{Verbatim} \end{minipage} -\begin{minipage}{.5\textwidth}\begin{tkzelements} +\begin{minipage}{.5\textwidth}\directlua{% +init_elements () z.O = point : new (0,0) z.A = point : new (1,2) C.OA = circle : new (z.O,z.A) @@ -1773,7 +1832,7 @@ BCn = C.OA : in_out (z.N) BDn = C.OA : in_out_disk (z.N) BCp = C.OA : in_out (z.P) BDp = C.OA : in_out_disk (z.P) -\end{tkzelements} +} \def\tkzPosPoint#1#2#3#4{% \tkzLabelPoints(O,M,N,P) \ifthenelse{\equal{\tkzUseLua{#1}}{true}}{ @@ -1835,7 +1894,8 @@ This function returns a string indicating the position of the circle in relation \begin{minipage}{.5\textwidth} \begin{Verbatim} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point : new ( 0 , 0 ) z.a = point : new ( 3 , 0 ) z.B = point : new ( 2 , 0 ) @@ -1846,7 +1906,7 @@ This function returns a string indicating the position of the circle in relation if position == "inside tangent" then color = "orange" else color = "blue" end -\end{tkzelements} +} \begin{tikzpicture} \tkzGetNodes @@ -1856,7 +1916,8 @@ This function returns a string indicating the position of the circle in relation \end{Verbatim} \end{minipage} \begin{minipage}{.5\textwidth} -\begin{tkzelements} +\directlua{% +init_elements () z.A = point : new ( 1 , 0 ) z.a = point : new ( 3 , 0 ) z.B = point : new ( 2 , 0 ) @@ -1865,7 +1926,7 @@ C.Aa = circle: new (z.A,z.a) C.Bb = circle: new (z.B,z.b) position = C.Aa : circles_position (C.Bb) if position == "inside tangent" then color = "orange" else color = "blue" end -\end{tkzelements} +} \hspace{\fill} \begin{tikzpicture} \tkzGetNodes |