Sangaku

376 /www/nrich/html/content/01/09/15plus4/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> The square $ABCD$ has sides of length 1 unit and it is split into three triangles by the lines $BP$ and $CP$. If $P$ is the midpoint of $AD$, find the radii of the inscribed circles of these triangles. Now suppose the lengths $AP$ and $PD$ are $(1- p)$ and $p$ respectively. Find the radii of the three circles $r_1$, $r_2$and $r_3$ in terms of $p$ and plot, on the same axes, the graphs of $r_1$, $r_2$ and $r_3$ as $p$ varies from 0 to 1. Can the ratio of the radii $r_1 : r_2 : r_3$ ever take the value $1:2:3$? <mdo:image alt="square ABCD with three circles within it." src="pic2.gif"></mdo:image> <div class="framework">Notes and Background Sangaku in Japanese means a mathematics tablet. During the Edo period (1603-1867) when Japan was cut off from the western world, people of all classes produced theorems in Euclidean geometry as beautifully coloured drawings on wooden tablets to be hung in a temple. Proofs were rarely given. The tablets challenged other geometers: "See if you can prove this."</div> </mdoxml> <mdoxml version="1.0"> This solution is from Etienne of Parramatta Highschool, NSW Australia I need to prove the area of a triangle $ A $ is given by $ A = rs $ where $ r $ is the radius of the incircle (inscribed circle) and $ s$ is the semi-perimeter (half the perimeter $ (a+b+c)/2$). Let $ a $, $ b $, $ c $ be the lengths of the sides of triangle $ ABC $. Join the incentre I of the triangle to the 3 corners. From I drop 3 perpendiculars to each of the sides, each of these has a length of $ r $. The areas of $ ABI $, $ BCI $, $ CAI $ are $ cr/2 $, $ ar/2 $ and $ br/2 $ respectively. They sum up to give the area of triangle $ ABC $ Area of $ ABC = cr/2 + ar/2 + br/2 = r(a+b+c)/2 = rs $. Back to the question! When $ P $ is the midpoint of $ AD, r_1 = r_2 $ because they sit on congruent triangles. The area of triangle $ ABP = AP \times AP/2 = 1/4. PB = \sqrt(1+1/4) = \sqrt5/2. $ The semi perimeter of $ ABP = (1 + 1/2 + \sqrt5/2)/2 = [3+\sqrt5]/4. $ Using the formula $ A = rs $ for the area of the triangle, the area of $ ABP = r_1[3+\sqrt5]/4 = 1/4. $ This gives $ r_1 = 1/[3 + \sqrt5] = [3 - \sqrt5]/4 $ so $ r_1 = r_2 = [3 - \sqrt5]/4. $ The area of BPC is 1 - 1/4 - 1/4 = 1/2 and $ PB = PC = \sqrt5/2. $ The semi-perimeter of $ BPC = [1+ \sqrt5/2 + \sqrt5/2]/2 = [1 + \sqrt5]/2. $ From the area of triangle $ BPC $ we get $ r_3[1 + \sqrt5]/2 = 1/2 $ so $ r_3 = 1/[1 + \sqrt5] = [\sqrt5 - 1]/4. $ Now suppose the lengths AP and PD are $ 1-p $ and $ p $. The area of $ APB = (1-p)/2. $ The length $ PB = \sqrt(1 + p^2 - 2p + 1) = \sqrt(p^2 - 2p + 2) $ and the semi perimeter of $ APB = [1 + (1-p) + \sqrt(p^2 - 2p + 2)]/2.$ So, using the area formula again, $$ r_2 = {1-p\over 2} \times {2\over [2 - p + \sqrt(p^2 - 2p + 2)]} = {1\over 2}[2 - p - \sqrt(p^2 - 2p + 2)]. $$ Similary with $ DPC $. The area of $ DPC = p/2 $ and the length $ PC = \sqrt(p^2 + 1). $ The semi perimeter of $ PC = [1 + p + \sqrt(p^2 + 1)]/2. $ So $$ r_1 = {p\over 2} \times {2\over [1 + p + \sqrt(p^2 + 1)]} = {1\over 2}[1 + p - \sqrt(p^2 + 1)]. $$ Similarly $$ r_3 = {1\over 2} \times {2\over [1 + \sqrt(p^2 + 1) + \sqrt(p^2 -2p + 2)]} = {1\over 2}[1 + \sqrt(p^2 + 1) - \sqrt(p^2 -2p +2)][\sqrt(p^2 +1)- p]. $$ To finish this question you need to use the formulae found above which give the radii $ r_1 $, $ r_2$ and $ r_3 $ in terms of $ p $ and then consider how these values change as $ p $ varies from $0$ to $1$. For example $ r_1 $ varies from $0$ to approximately $0.293$ as $ p $ increases from $0$ to $1$. In order to discover whether the ratio of the radii $ r_1 : r_2 : r_3 $ can ever take the value $1 : 2 : 3$ you could plot on the same axes the graphs of $ r_1 $, $ r_2 $ and $ r_3 $ as $ p $ varies from $0$ to $1$. </mdoxml> 2 3 0 0 0 0 1 Sangaku The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD. Measures and Mensuration Area 2D Geometry, Shape and Space Inscribed circle 2D Geometry, Shape and Space Radius (radii) & diameters Sequences, Functions and Graphs Graphs