\newpage \section{Circles} Among the following macros, one will allow you to draw a circle, which is not a real feat. To do this, you will need to know the center of the circle and either the radius of the circle or a point on the circumference. It seemed to me that the most frequent use was to draw a circle with a given center passing through a given point. This will be the default method, otherwise you will have to use the \tkzname{R} option. There are a large number of special circles, for example the circle circumscribed by a triangle. \begin{itemize} \item I have created a first macro \tkzcname{tkzDefCircle} which allows, according to a particular circle, to retrieve its center and the measurement of the radius in cm. This recovery is done with the macros \tkzcname{tkzGetPoint} and \tkzcname{tkzGetLength}; \item then a macro \tkzcname{tkzDrawCircle}; \item then a macro that allows you to color in a disc, but without drawing the circle \tkzcname{tkzFillCircle}; \item sometimes, it is necessary for a drawing to be contained in a disk, this is the role assigned to \tkzcname{tkzClipCircle}; \item it finally remains to be able to give a label to designate a circle and if several possibilities are offered, we will see here \tkzcname{tkzLabelCircle}. \end{itemize} \subsection{Characteristics of a circle: \tkzcname{tkzDefCircle}} This macro allows you to retrieve the characteristics (center and radius) of certain circles. \begin{NewMacroBox}{tkzDefCircle}{\oarg{local options}\parg{A,B} or \parg{A,B,C}}% \tkzHandBomb\ Attention the arguments are lists of two or three points. This macro is either used in partnership with \\ \tkzcname{tkzGetPoints} to obtain the center and a point on the circle, or by using \\ \tkzname{tkzFirstPointResult} and \tkzname{tkzSecondPointResult} if it is not necessary to keep the results. You can also use \tkzcname{tkzGetLength} to get the radius. \medskip \begin{tabular}{lll}% \toprule arguments & example & explanation \\ \midrule \TAline{\parg{pt1,pt2} or \parg{pt1,pt2,pt3}}{\parg{A,B}} {$[AB]$ is radius $A$ is the center} \bottomrule \end{tabular} \medskip \begin{tabular}{lll}% \toprule options & default & definition \\ \midrule \TOline{R} {circum}{circle characterized by a center and a radius} \TOline{diameter}{circum}{circle characterized by two points defining a diameter} \TOline{circum} {circum}{circle circumscribed of a triangle} \TOline{in} {circum}{incircle a triangle} \TOline{ex} {circum}{excircle of a triangle} \TOline{euler or nine}{circum}{Euler's Circle} \TOline{spieker} {circum}{Spieker Circle} \TOline{apollonius} {circum}{circle of Apollonius} \TOline{orthogonal from} {circum}{[orthogonal from = A ](O,M)} \TOline{orthogonal through}{circum}{[orthogonal through = A and B](O,M)} \TOline{K} {1}{coefficient used for a circle of Apollonius} \bottomrule \end{tabular} \medskip \emph{In the following examples, I draw the circles with a macro not yet presented. You may only need the center and a point on the circle. } \end{NewMacroBox} \subsubsection{Example with option \tkzname{R}} We obtain with the macro \tkzcname{tkzGetPoint} a point of the circle which is the East pole. \begin{tkzexample}[latex=7cm,small] \begin{tikzpicture}[scale=1] \tkzDefPoint(3,3){C} \tkzDefPoint(5,5){A} \tkzCalcLength(A,C) \tkzGetLength{rAC} \tkzDefCircle[R](C,\rAC) \tkzGetPoint{B} \tkzDrawCircle(C,B) \tkzDrawSegment(C,A) \tkzLabelSegment[above left](C,A){$2\sqrt{2}$} \tkzDrawPoints(A,B,C) \tkzLabelPoints(A,C,B) \end{tikzpicture} \end{tkzexample} \subsubsection{Example with option \tkzname{diameter}} It is simpler here to search directly for the middle of $[AB]$. The result is the center and if necessary \begin{tkzexample}[latex=7cm,small] \begin{tikzpicture} \tkzDefPoint(0,0){O} \tkzDefPoint(2,2){B} \tkzDefCircle[diameter](O,B) \tkzGetPoint{A} \tkzDrawCircle(A,B) \tkzDrawPoints(O,A,B) \tkzDrawSegment(O,B) \tkzLabelPoints(O,A,B) \tkzLabelSegment[above left](O,A){$\sqrt{2}$} \tkzLabelSegment[above left](A,B){$\sqrt{2}$} \tkzMarkSegments[mark=s||](O,A A,B) \end{tikzpicture} \end{tkzexample} \subsubsection{Circles inscribed and circumscribed for a given triangle} \begin{tkzexample}[latex=7cm,small] \begin{tikzpicture}[scale=.75] \tkzDefPoint(2,2){A} \tkzDefPoint(5,-2){B} \tkzDefPoint(1,-2){C} \tkzDefCircle[in](A,B,C) \tkzGetPoints{I}{x} \tkzDefCircle[circum](A,B,C) \tkzGetPoint{K} \tkzDrawCircles[new](I,x K,A) \tkzLabelPoints[below](B,C) \tkzLabelPoints[above left](A,I,K) \tkzDrawPolygon(A,B,C) \tkzDrawPoints(A,B,C,I,K) \end{tikzpicture} \end{tkzexample} \subsubsection{Example with option \tkzname{ex}} We want to define an excircle of a triangle relatively to point $C$ \begin{tkzexample}[latex=8cm,small] \begin{tikzpicture}[scale=.75] \tkzDefPoints{ 0/0/A,4/0/B,0.8/4/C} \tkzDefCircle[ex](B,C,A) \tkzGetPoints{J_c}{h} \tkzDefPointBy[projection=onto A--C ](J_c) \tkzGetPoint{X_c} \tkzDefPointBy[projection=onto A--B ](J_c) \tkzGetPoint{Y_c} \tkzDefCircle[in](A,B,C) \tkzGetPoints{I}{y} \tkzDrawCircles[color=lightgray](J_c,h I,y) \tkzDefPointBy[projection=onto A--C ](I) \tkzGetPoint{F} \tkzDefPointBy[projection=onto A--B ](I) \tkzGetPoint{D} \tkzDrawPolygon(A,B,C) \tkzDrawLines[add=0 and 1.5](C,A C,B) \tkzDrawSegments(J_c,X_c I,D I,F J_c,Y_c) \tkzMarkRightAngles(A,F,I B,D,I J_c,X_c,A J_c,Y_c,B) \tkzDrawPoints(B,C,A,I,D,F,X_c,J_c,Y_c) \tkzLabelPoints(B,A,J_c,I,D) \tkzLabelPoints[above](Y_c) \tkzLabelPoints[left](X_c) \tkzLabelPoints[above left](C) \tkzLabelPoints[left](F) \end{tikzpicture} \end{tkzexample} \subsubsection{Euler's circle for a given triangle with option \tkzname{euler}} We verify that this circle passes through the middle of each side. \begin{tkzexample}[latex=6cm,small] \begin{tikzpicture}[scale=.75] \tkzDefPoint(5,3.5){A} \tkzDefPoint(0,0){B} \tkzDefPoint(7,0){C} \tkzDefCircle[euler](A,B,C) \tkzGetPoints{E}{e} \tkzDefSpcTriangle[medial](A,B,C){M_a,M_b,M_c} \tkzDrawCircle[new](E,e) \tkzDrawPoints(A,B,C,E,M_a,M_b,M_c) \tkzDrawPolygon(A,B,C) \tkzLabelPoints[below](B,C) \tkzLabelPoints[left](A,E) \end{tikzpicture} \end{tkzexample} \subsubsection{Apollonius circles for a given segment option \tkzname{apollonius}} \begin{tkzexample}[latex=9cm,small] \begin{tikzpicture}[scale=0.75] \tkzDefPoint(0,0){A} \tkzDefPoint(4,0){B} \tkzDefCircle[apollonius,K=2](A,B) \tkzGetPoints{K1}{x} \tkzDrawCircle[color = teal!50!black, fill=teal!20,opacity=.4](K1,x) \tkzDefCircle[apollonius,K=3](A,B) \tkzGetPoints{K2}{y} \tkzDrawCircle[color=orange!50, fill=orange!20,opacity=.4](K2,y) \tkzLabelPoints[below](A,B,K1,K2) \tkzDrawPoints(A,B,K1,K2) \tkzDrawLine[add=.2 and 1](A,B) \end{tikzpicture} \end{tkzexample} \subsubsection{Circles exinscribed to a given triangle option \tkzname{ex}} You can also get the center and the projection of it on one side of the triangle. with \tkzcname{tkzGetFirstPoint\{Jb\}} and \tkzcname{tkzGetSecondPoint\{Tb\}}. \begin{tkzexample}[latex=8cm,small] \begin{tikzpicture}[scale=.6] \tkzDefPoint(0,0){A} \tkzDefPoint(3,0){B} \tkzDefPoint(1,2.5){C} \tkzDefCircle[ex](A,B,C) \tkzGetPoints{I}{i} \tkzDefCircle[ex](C,A,B) \tkzGetPoints{J}{j} \tkzDefCircle[ex](B,C,A) \tkzGetPoints{K}{k} \tkzDefCircle[in](B,C,A) \tkzGetPoints{O}{o} \tkzDrawCircles[new](J,j I,i K,k O,o) \tkzDrawLines[add=1.5 and 1.5](A,B A,C B,C) \tkzDrawPolygon[purple](I,J,K) \tkzDrawSegments[new](A,K B,J C,I) \tkzDrawPoints(A,B,C) \tkzDrawPoints[new](I,J,K) \tkzLabelPoints(A,B,C,I,J,K) \end{tikzpicture} \end{tkzexample} \subsubsection{Spieker circle with option \tkzname{spieker}} The incircle of the medial triangle $M_aM_bM_c$ is the Spieker circle: \begin{tkzexample}[latex=6cm, small] \begin{tikzpicture}[scale=1] \tkzDefPoints{ 0/0/A,4/0/B,0.8/4/C} \tkzDefSpcTriangle[medial](A,B,C){M_a,M_b,M_c} \tkzDefTriangleCenter[spieker](A,B,C) \tkzGetPoint{S_p} \tkzDrawPolygon(A,B,C) \tkzDrawPolygon[cyan](M_a,M_b,M_c) \tkzDrawPoints(B,C,A) \tkzDefCircle[spieker](A,B,C) \tkzDrawPoints[new](M_a,M_b,M_c,S_p) \tkzDrawCircle[new](tkzFirstPointResult,tkzSecondPointResult) \tkzLabelPoints[right](M_a) \tkzLabelPoints[left](M_b) \tkzLabelPoints[below](A,B,M_c,S_p) \tkzLabelPoints[above](C) \end{tikzpicture} \end{tkzexample} \subsection{Projection of excenters} \begin{NewMacroBox}{tkzDefProjExcenter}{\oarg{local options}\parg{A,B,C}\parg{a,b,c}\marg{X,Y,Z}}% Each excenter has three projections on the sides of the triangle ABC. We can do this with one macro\\ \tkzcname{tkzDefProjExcenter[name=J](A,B,C)(a,b,c)\{Y,Z,X\}}. \medskip \begin{tabular}{lll}% \toprule options & default & definition \\ \midrule \TOline{name} {no defaut}{used to name the vertices} \bottomrule \end{tabular} \begin{tabular}{lll}% arguments & default & definition \\ \midrule \TAline{(pt1=$\alpha_1$,pt2=$\alpha_2$,\dots)}{no default}{Each point has a assigned weight} \end{tabular} \medskip \end{NewMacroBox} \subsubsection{\tkzname{Excircles}} \begin{tkzexample}[vbox,small] \begin{tikzpicture}[scale=.6] \tikzset{line style/.append style={line width=.2pt}} \tikzset{label style/.append style={color=teal,font=\footnotesize}} \tkzDefPoints{0/0/A,5/0/B,0.8/4/C} \tkzDefSpcTriangle[excentral,name=J](A,B,C){a,b,c} \tkzDefSpcTriangle[intouch,name=I](A,B,C){a,b,c} \tkzDefProjExcenter[name=J](A,B,C)(a,b,c){X,Y,Z} \tkzDefCircle[in](A,B,C) \tkzGetPoint{I} \tkzGetSecondPoint{T} \tkzDrawCircles[red](Ja,Xa Jb,Yb Jc,Zc) \tkzDrawCircle(I,T) \tkzDrawPolygon[dashed,color=blue](Ja,Jb,Jc) \tkzDrawLines[add=1.5 and 1.5](A,C A,B B,C) \tkzDrawSegments(Ja,Xa Ja,Ya Ja,Za Jb,Xb Jb,Yb Jb,Zb Jc,Xc Jc,Yc Jc,Zc I,Ia I,Ib I,Ic) \tkzMarkRightAngles[size=.2,fill=gray!15](Ja,Za,B Ja,Xa,B Ja,Ya,C Jb,Yb,C) \tkzMarkRightAngles[size=.2,fill=gray!15](Jb,Zb,B Jb,Xb,C Jc,Yc,A Jc,Zc,B Jc,Xc,C I,Ia,B I,Ib,C I,Ic,A) \tkzDrawSegments[blue](Jc,C Ja,A Jb,B) \tkzDrawPoints(A,B,C,Xa,Xb,Xc,Ja,Jb,Jc,Ia,Ib,Ic,Ya,Yb,Yc,Za,Zb,Zc) \tkzLabelPoints(A,Ya,Yb,Ja,I) \tkzLabelPoints[left](Jb,Ib,Yc) \tkzLabelPoints[below](Zb,Ic,Jc,B,Za,Xa) \tkzLabelPoints[above right](C,Zc,Yb) \tkzLabelPoints[right](Xb,Ia,Xc) \end{tikzpicture} \end{tkzexample} \subsubsection{\tkzname{Orthogonal from}} Orthogonal circle of given center. \tkzcname{tkzGetPoints\{z1\}\{z2\}} gives two points of the circle. \begin{tkzexample}[latex=7cm,small] \begin{tikzpicture}[scale=.75] \tkzDefPoints{0/0/O,1/0/A} \tkzDefPoints{1.5/1.25/B,-2/-3/C} \tkzDefCircle[orthogonal from=B](O,A) \tkzGetPoints{z1}{z2} \tkzDefCircle[orthogonal from=C](O,A) \tkzGetPoints{t1}{t2} \tkzDrawCircle(O,A) \tkzDrawCircles[new](B,z1 C,t1) \tkzDrawPoints(t1,t2,C) \tkzDrawPoints(z1,z2,O,A,B) \tkzLabelPoints[right](O,A,B,C) \end{tikzpicture} \end{tkzexample} \subsubsection{\tkzname{Orthogonal through}} Orthogonal circle passing through two given points. \begin{tkzexample}[latex=6cm,small] \begin{tikzpicture}[scale=1] \tkzDefPoint(0,0){O} \tkzDefPoint(1,0){A} \tkzDrawCircle(O,A) \tkzDefPoint(-1.5,-1.5){z1} \tkzDefPoint(1.5,-1.25){z2} \tkzDefCircle[orthogonal through=z1 and z2](O,A) \tkzGetPoint{c} \tkzDrawCircle[new](tkzPointResult,z1) \tkzDrawPoints[new](O,A,z1,z2,c) \tkzLabelPoints[right](O,A,z1,z2,c) \end{tikzpicture} \end{tkzexample} \endinput