The following code is compiled by TikZ to draw a right triangle so that the enclosed area is partitioned by infinitely many triangles similar to itself. I saw on this site an image of this from an old textbook. The post indicated that the display is a geometric interpretation of the geometric series. I would like to either find that post or know how this display gives a geometric interpretation of the geometric series.
\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\noindent \hspace*{\fill}
\begin{tikzpicture}
%Rays k and $\ell$ have a common endpoint at the origin, which is labeled A. With respect to
%the horizontal line through A, k is inclined at an angle of 50 degrees, and $\ell$ is
%inclined at an angle of 15 degrees. To place the label for A, the rays are extended 15pt by an
%invisible path command "before" A. The endpoints of these extensions are called "label_A_below"
%and "label_A_above" and the midpoint of the line segment between label_A_below and label_A_above
%is called "label_A". A node command typesets "A" at the point 7.5pt from A on the invisible
%line segment between A and label_A .
\coordinate (A) at (0,0);
\draw[fill] (A) circle (1.5pt);
%B' is a point on ray k and C' is a point on $\ell$. These points are used to draw two
%invisible paths from A. On these paths, two sides of a triangle will be drawn.
\coordinate (B') at (50:9.25);
\path[name path=ray_k] (A) -- (B');
\coordinate (C') at (15:7.5);
\path[name path=ray_ell] (A) -- (C');
\path (A) -- (C');
%These commands label a point "B_3" 3.75cm from A on ray $k$. (Two more points B_2 and B_1 on k
%that are further from A than B_3 are to be defined, and three more points B_4, B_5, and B_6
%on k that are closer to A than B_3 are to be defined.)
\coordinate (B_3) at (50:{15/4});
%These commands label the projection of AB_3 onto \ell "C_3." A line segment is drawn between
%B_3 and C_3, and a right-angle mark is placed at C_3. (Commands for the label for C_3 are
%issued after the point $B_{4}$ on k is defined.)
\coordinate (C_3) at ($(A)!(B_3)!(C')$);
\draw (B_3) -- (C_3);
%The following command make the right-angle mark at C_3.
\coordinate (U_3) at ($(C_3)!3mm!45:(C')$);
\draw (U_3) -- ($(C_3)!(U_3)!(C')$);
\draw (U_3) -- ($(C_3)!(U_3)!(B_3)$);
%These commands draw a line segment perpendicular to ray k from B_3 to ray $\ell$.
%The intersection is labeled C_2. (The label for C_2 is placed after the point
%B_2 is defined.)
\path[name path=perpendicular_line_segment_from_B_3] (B_3) -- ($(B_3)!3.5cm!90:(A)$);
\coordinate[name intersections={of=perpendicular_line_segment_from_B_3 and ray_ell,by={C_2}}];
\draw (B_3) -- (C_2);
%The following commands make the right-angle mark at B_3.
\coordinate (V_3) at ($(B_3)!3mm!-45:(B')$);
\draw (V_3) -- ($(B_3)!(V_3)!(B')$);
\draw (V_3) -- ($(B_3)!(V_3)!(C_2)$);
%The following commands define B_2 as the intersection of k and a line segment
%perpendicular to $\ell$ through C_2.
\path[name path=perpendicular_line_segment_from_C_2] (C_2) -- ($(C_2)!4cm!90:(C')$);
\coordinate[name intersections={of=perpendicular_line_segment_from_C_2 and ray_k,by={B_2}}];
\draw (C_2) -- (B_2);
%The following commands make the right-angle mark at C_2.
\coordinate (U_2) at ($(C_2)!3mm!45:(C')$);
\draw (U_2) -- ($(C_2)!(U_2)!(C')$);
\draw (U_2) -- ($(C_2)!(U_2)!(B_2)$);
%These commands draw a line segment perpendicular to ray k from B_2 to ray $\ell$.
%The intersection is labeled C_1. (The label for C_1 is placed after the point
%B_1 is defined.)
\path[name path=perpendicular_line_segment_from_B_2] (B_2) -- ($(B_2)!5cm!90:(A)$);
\coordinate[name intersections={of=perpendicular_line_segment_from_B_2 and ray_ell,by={C_1}}];
\draw (B_2) -- (C_1);
%The following commands make the right-angle mark at B_1.
\coordinate (V_2) at ($(B_2)!3mm!-45:(B')$);
\draw (V_2) -- ($(B_2)!(V_2)!(B')$);
\draw (V_2) -- ($(B_2)!(V_2)!(C_1)$);
%The following commands define B_1 as the intersection of k and a line segment
%perpendicular to $\ell$ through C_1.
\path[name path=line_segment_from_C_1_to_ray_k] (C_1) -- ($(C_1)!5.25cm!90:(C')$);
\coordinate[name intersections={of=line_segment_from_C_1_to_ray_k and ray_k,by={B_1}}];
\draw (C_1) -- (B_1);
%These commands label the projection of AC_3 onto k "B_4."
\coordinate (B_4) at ($(A)!(C_3)!(B')$);
\draw (B_4) -- (C_3);
%These commands label the projection of AB_4 onto $\ell$ "C_4."
\coordinate (C_4) at ($(A)!(B_4)!(C')$);
\draw (B_4) -- (C_4);
%The following commands make the right-angle mark at B_4.
\coordinate (V_4) at ($(B_4)!3mm!-45:(B')$);
\draw (V_4) -- ($(B_4)!(V_4)!(B')$);
\draw (V_4) -- ($(B_4)!(V_4)!(C_3)$);
%These commands label the projection of AC_4 onto k "B_5."
\coordinate (B_5) at ($(A)!(C_4)!(B')$);
\draw (B_5) -- (C_4);
%The following commands make the right-angle mark at C_4.
\coordinate (U_4) at ($(C_4)!3mm!45:(C')$);
\draw (U_4) -- ($(C_4)!(U_4)!(C')$);
\draw (U_4) -- ($(C_4)!(U_4)!(B_4)$);
%The following commands make the right-angle mark at B_5.
\coordinate (V_5) at ($(B_5)!3mm!-45:(B')$);
\draw (V_5) -- ($(B_5)!(V_5)!(B')$);
\draw (V_5) -- ($(B_5)!(V_5)!(C_4)$);
%These commands label the projection of AB_5 onto $\ell$ "C_5."
\coordinate (C_5) at ($(A)!(B_5)!(C')$);
\draw (B_5) -- (C_5);
%The following commands make the right-angle mark at C_5.
\coordinate (U_5) at ($(C_5)!3mm!45:(C')$);
\draw (U_5) -- ($(C_5)!(U_5)!(C')$);
\draw (U_5) -- ($(C_5)!(U_5)!(B_5)$);
%The following commands draw five more smaller, similar triangles, with dashed lines,
%to suggest that the process continues.
\coordinate (B_6) at ($(A)!(C_5)!(B')$);
\draw[dashed] (C_5) -- (B_6);
\coordinate (C_6) at ($(A)!(B_6)!(C')$);
\draw[dashed] (B_6) -- (C_6);
\coordinate (B_7) at ($(A)!(C_6)!(B')$);
\draw[dash dot] (C_6) -- (B_7);
\coordinate (C_7) at ($(A)!(B_7)!(C')$);
\draw[dash dot] (B_7) -- (C_7);
\coordinate (B_8) at ($(A)!(C_7)!(B')$);
\draw[dotted] (C_7) -- (B_8);
\coordinate (C_8) at ($(A)!(B_8)!(C')$);
\draw[dotted] (B_8) -- (C_8);
%These commands draw the two sides of the triangle with the common endpoint A.
\draw (A) -- (C_1);
\draw (A) -- (B_1);
\end{tikzpicture}
\end{document}
