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tkz-euclide

Lines, triangles and polygons

14 Straight lines

Drawing straight lines is of course essential, but first we must be able to define particular lines such as mediators (perpendicular bisectors), bisectors, parallels and perpendiculars. The principle is to determine two points of the line.

14.1 Definition of straight lines

  • \tkzDefLine[⟨local options⟩](⟨pt1,pt2⟩) or (⟨pt1,pt2,pt3⟩)

    • The argument is a list of two or three points. Depending on the case, the macro defines one or two points needed to obtain the line; retrieve them with \tkzGetPoint or \tkzGetPoints. I use the term "mediator" for the perpendicular bisector of a segment.

    argumentsexampleexplanation
    (⟨pt1,pt2⟩)[mediator](⟨A,B⟩)mediator of the segment \([A,B]\)
    (⟨pt1,pt2,pt3⟩)[bisector](⟨A,B,C⟩)bisector of \(\widehat{ABC}\)
    (⟨pt1,pt2,pt3⟩)[altitude](⟨A,B,C⟩)altitude from \(B\)
    (⟨pt1⟩)[tangent at=A](⟨O⟩)tangent at \(A\) on the circle center \(O\)
    (⟨pt1,pt2⟩)[tangent from=A](⟨O,B⟩)circle center \(O\) through \(B\)
    optionsdefaultdefinition
    mediatorperpendicular bisector of a line segment
    perpendicular=through…mediatorperpendicular to a line passing through a point
    orthogonal=through…mediatorsee above
    parallel=through…mediatorparallel to a line passing through a point
    bisectormediatorbisector of an angle defined by three points
    bisector outmediatorexterior angle bisector
    symmedianmediatorsymmedian from a vertex
    altitudemediatoraltitude from avertex
    eulermediatoreuler line of a triangle
    tangent atmediatortangent at a point of a circle
    tangent frommediatortangent from an exterior point
    K1coefficient for the perpendicular line
    normedfalsenormalizes the created segment
14.1.1 With mediator

Compiled output

\begin{tikzpicture}[rotate=25]
\tkzDefPoints{-2/0/A,1/2/B}
\tkzDefLine[mediator](A,B) \tkzGetPoints{C}{D}
\tkzDefPointWith[linear,K=.75](C,D) \tkzGetPoint{D}
\tkzDefMidPoint(A,B) \tkzGetPoint{I}
\tkzFillPolygon[color=teal!20](A,C,B,D)
\tkzDrawSegments(A,B C,D)
\tkzMarkRightAngle(B,I,C)
\tkzDrawSegments(D,B D,A)
\tkzDrawSegments(C,B C,A)
\end{tikzpicture}
14.1.2 An envelope with option mediator

Based on a figure by O. Reboux, made with pst-eucl by D. Rodriguez.

Compiled output

\begin{tikzpicture}[scale=.75]
% necessary
\tkzInit[xmin=-6,ymin=-4,xmax=6,ymax=6]
\tkzClip
\tkzSetUpLine[thin,color=magenta]
\tkzDefPoint(0,0){O}
\tkzDefPoint(132:4){A}
\tkzDefPoint(5,0){B}
\foreach \ang in {5,10,...,360}{%
\tkzDefPoint(\ang:5){M}
\tkzDefLine[mediator](A,M)
\tkzGetPoints{x}{y}
\tkzDrawLine[add= 3 and 3](x,y)}
\end{tikzpicture}
14.1.3 A parabola with option mediator

Based on a figure by O. Reboux, made with pst-eucl by D. Rodriguez. It is not necessary to name the two points that define the mediator.

Compiled output

\begin{tikzpicture}[scale=.6]
\tkzInit[xmin=-6,ymin=-4,xmax=6,ymax=6]
\tkzClip
\tkzSetUpLine[thin,color=teal]
\tkzDefPoint(0,0){O}
\tkzDefPoint(132:5){A}
\tkzDefPoint(4,0){B}
\foreach \ang in {5,10,...,360}{%
\tkzDefPoint(\ang:4){M}
\tkzDefLine[mediator](A,M)
\tkzGetPoints{x}{y}
\tkzDrawLine[add= 3 and 3](x,y)}
\end{tikzpicture}
14.1.4 With options bisector and normed

Compiled output

\begin{tikzpicture}[rotate=25,scale=.75]
\tkzDefPoints{0/0/C, 2/-3/A, 4/0/B}
\tkzDefLine[bisector,normed](B,A,C) \tkzGetPoint{a}
\tkzDrawLines[add= 0 and .5](A,B A,C)
\tkzShowLine[bisector,gap=4,size=2,color=red](B,A,C)
\tkzDrawLines[new,dashed,add= 0 and 3](A,a)
\end{tikzpicture}
14.1.5 With option parallel=through

Archimedes’ Book of Lemmas, Proposition 1.

Compiled output

\begin{tikzpicture}
\tkzDefPoints{0/0/O_1,0/1/O_2,0/3/A}
\tkzDefPoint(15:3){F}
\tkzDefPointBy[symmetry=center O_1](F)
\tkzGetPoint{E}
\tkzDefLine[parallel=through O_2](E,F)
\tkzGetPoint{x}
\tkzInterLC(x,O_2)(O_2,A) \tkzGetPoints{D}{C}
\tkzDrawCircles(O_1,A O_2,A)
\tkzDrawSegments[new](O_1,A E,F C,D)
\tkzDrawSegments[purple](A,E A,F)
\tkzDrawPoints(A,O_1,O_2,E,F,C,D)
\tkzLabelPoints(A,O_1,O_2,E,F,C,D)
\end{tikzpicture}
14.1.6 With option orthogonal and parallel

Compiled output

\begin{tikzpicture}
\tkzDefPoints{-1.5/-0.25/A,1/-0.75/B,-0.7/1/C}
\tkzDrawLine(A,B)
\tkzLabelLine[pos=1.25,below left](A,B){$(d_1)$}
\tkzDrawPoints(A,B,C)
\tkzDefLine[orthogonal=through C](B,A) \tkzGetPoint{c}
\tkzDrawLine(C,c)
\tkzLabelLine[pos=1.25,left](C,c){$(\delta)$}
\tkzInterLL(A,B)(C,c) \tkzGetPoint{I}
\tkzMarkRightAngle(C,I,B)
\tkzDefLine[parallel=through C](A,B) \tkzGetPoint{c'}
\tkzDrawLine(C,c')
\tkzLabelLine[pos=1.25,below left](C,c'){$(d_2)$}
\tkzMarkRightAngle(I,C,c')
\end{tikzpicture}
14.1.7 With option altitude

Compiled output

\begin{tikzpicture}
\tkzDefPoints{0/0/A,6/0/B,0.8/4/C}
\tkzDefLine[altitude](A,B,C) \tkzGetPoint{b}
\tkzDefLine[altitude](B,C,A) \tkzGetPoint{c}
\tkzDefLine[altitude](B,A,C) \tkzGetPoint{a}
\tkzDrawPolygon(A,B,C)
\tkzDrawPoints[blue](a,b,c)
\tkzDrawSegments[blue](A,a B,b C,c)
\tkzLabelPoints(A,B,c)
\tkzLabelPoints[above](C,a)
\tkzLabelPoints[above left](b)
\end{tikzpicture}
14.1.8 With option euler

Compiled output

\begin{tikzpicture}[scale=.75]
\tkzDefPoints{0/0/A,6/0/B,0.8/4/C}
\tkzDefLine[euler](A,B,C)
\tkzGetPoints{h}{e}
\tkzDefTriangleCenter[circum](A,B,C)
\tkzGetPoint{o}
\tkzDrawPolygon[teal](A,B,C)
\tkzDrawPoints[red](A,B,C,h,e,o)
\tkzDrawLine[add= 2 and 2](h,e)
\tkzLabelPoints(A,B,C,h,e,o)
\tkzLabelPoints[above](C)
\end{tikzpicture}
14.1.9 Tangent passing through a point on the circle tangent at

Compiled output

\begin{tikzpicture}[scale=.75]
\tkzDefPoint(0,0){O}
\tkzDefPoint(6,6){E}
\tkzDefRandPointOn[circle=center O radius 3]
\tkzGetPoint{A}
\tkzDrawSegment(O,A)
\tkzDrawCircle(O,A)
\tkzDefLine[tangent at=A](O)
\tkzGetPoint{h}
\tkzDrawLine[add = 4 and 3](A,h)
\tkzMarkRightAngle[fill=teal!30](O,A,h)
\end{tikzpicture}
14.1.10 Choice of contact point with tangents passing through an external point option tangent from

The tangent is not drawn. With the option at, a point of the tangent is given by tkzPointResult. With the option from, you get two points of the circle, tkzFirstPointResult and tkzSecondPointResult. You can choose between them by comparing the angles formed by the outer point, the contact point and the center: the two angles have opposite orientations, and the counterclockwise one corresponds to tkzFirstPointResult.

Compiled output

\begin{tikzpicture}[scale=1,rotate=-30]
\tkzDefPoints{0/0/Q,0/2/A,6/-1/O}
\tkzDefLine[tangent from = O](Q,A)
\tkzGetPoints{R}{S}
\tkzInterLC[near](O,Q)(Q,A)
\tkzGetPoints{M}{N}
\tkzDrawCircle(Q,M)
\tkzDrawSegments[new,add = 0 and .2](O,R O,S)
\tkzDrawSegments[gray](N,O R,Q S,Q)
\tkzDrawPoints(O,Q,R,S,M,N)
\tkzMarkAngle[gray,-stealth,size=1](O,R,Q)
\tkzFindAngle(O,R,Q) \tkzGetAngle{an}
\tkzLabelAngle(O,R,Q){%
$\pgfmathprintnumber{\an}^\circ$}
\tkzMarkAngle[gray,-stealth,size=1](O,S,Q)
\tkzFindAngle(O,S,Q) \tkzGetAngle{an}
\tkzLabelAngle(O,S,Q){%
$\pgfmathprintnumber{\an}^\circ$}
\tkzLabelPoints(Q,O,M,N,R)
\tkzLabelPoints[above,text=red](S)
\end{tikzpicture}
14.1.11 Example of tangents passing through an external point

Compiled output

\begin{tikzpicture}[scale=.8]
\tkzDefPoints{0/0/c,1/0/d,3/0/a0}
\def\tkzRadius{1}
\tkzDrawCircle(c,d)
\foreach \an in {0,10,...,350}{
\tkzDefPointBy[rotation=center c angle \an](a0)
\tkzGetPoint{a}
\tkzDefLine[tangent from = a](c,d)
\tkzGetPoints{e}{f}
\tkzDrawLines(a,f a,e)
\tkzDrawSegments(c,e c,f)}
\end{tikzpicture}
14.1.12 Example of Andrew Mertz

Compiled output

\begin{tikzpicture}[scale=.6]
\tkzDefPoint(100:8){A}\tkzDefPoint(50:8){B}
\tkzDefPoint(0,0){C} \tkzDefPoint(0,-4){R}
\tkzDrawCircle(C,R)
\tkzDefLine[tangent from = A](C,R) \tkzGetPoints{D}{E}
\tkzDefLine[tangent from = B](C,R) \tkzGetPoints{F}{G}
\tkzDrawSector[fill=teal!20,opacity=0.5](A,E)(D)
\tkzFillSector[color=teal,opacity=0.5](B,G)(F)
\end{tikzpicture}

http://www.texample.net/tikz/examples/

14.1.13 Drawing a tangent option tangent from

Compiled output

\begin{tikzpicture}[scale=.6]
\tkzDefPoint(0,0){B}
\tkzDefPoint(0,8){A}
\tkzDefSquare(A,B)
\tkzGetPoints{C}{D}
\tkzDrawPolygon(A,B,C,D)
\tkzClipPolygon(A,B,C,D)
\tkzDefPoint(4,8){F}
\tkzDefPoint(4,0){E}
\tkzDefPoint(4,4){Q}
\tkzFillPolygon[color = green](A,B,C,D)
\tkzDrawCircle[fill = orange](B,A)
\tkzDrawCircle[fill = purple](E,B)
\tkzDefLine[tangent from = B](F,A)
\tkzInterLL(F,tkzSecondPointResult)(C,D)
\tkzInterLL(A,tkzPointResult)(F,E)
\tkzDrawCircle[fill = yellow](tkzPointResult,Q)
\tkzDefPointBy[projection= onto B--A](tkzPointResult)
\tkzDrawCircle[fill = blue!50!black](tkzPointResult,A)
\end{tikzpicture}