Let $ABC$ be a right angled triangle at $A$. Denote $D$ the foot of the altitude through $A$ and $O_1, O_2$ the incentres of triangles $ADB$ and $ADC$. The circle with centre $A$ and radius $AD$ cuts $AB$ in $K$ and $AC$ in $L$. Show that $O_1, O_2, K$ and $L$ are on a line.
Problem
Source: Pan African MO 2006 Q6
Tags: symmetry, geometry, geometric transformation, rotation, Asymptote, geometry unsolved
leader
01.05.2013 15:54
Spiral symmetry centered at $D$ that takes $O_{1}$ to $O_{2}$ and $B$ to $A$ has a rotation with angle $45$. so if $O_{1}O_{2}$ meets $AB,AC$ at $P,Q$ then $\angle APQ=45=\angle AQP=\angle AQO_{1}=\angle ADO_{1}=\angle ADO_{2}=\angle APO_{2}$ so $AP=AD=AQ$. I'm pretty sure this was discussed before.
Particle
01.05.2013 17:05
Since $CD$ is a tangent, $\angle CDL=\frac 1 2 \angle DAL=\frac 1 2 \angle B$. Since $\angle CDL+\angle LDO_2=45^\circ$, we get $\angle LDO_2=\frac 1 2 \angle C=\angle LCO_2$. Therefore $C,L,O_2,D$ concyclic and thus $\angle ALO_2=\angle CDO_2=45^\circ=\angle ALK\implies \; O_2\in KL$ and similarly $O_1\in KL$. [asy][asy]/* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(8cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 0.21, xmax = 16.62, ymin = -3.22, ymax = 4.74; /* image dimensions */ /* draw figures */ draw(shift((2.92,0.24)) * scale(2.54, 2.54)*unitcircle); draw((2.89,4.41)--(2.92,0.24)); draw((2.92,0.24)--(6.12,0.26)); draw((2.89,4.41)--(6.12,0.26)); draw((2.92,0.24)--(4.92,1.8)); draw((3.75,1.95)--(2.92,0.24)); draw((4.92,1.8)--(3.75,1.95)); draw((2.9,2.78)--(4.92,1.8)); draw((2.89,4.41)--(3.75,1.95)); draw((2.9,2.78)--(3.75,1.95)); /* dots and labels */ dot((2.92,0.24),dotstyle); label("$A$", (2.97,0.32), NE * labelscalefactor); dot((6.12,0.26),dotstyle); label("$B$", (6.17,0.34), NE * labelscalefactor); dot((2.89,4.41),dotstyle); label("$C$", (2.95,4.49), NE * labelscalefactor); dot((4.92,1.8),dotstyle); label("$D$", (4.98,1.88), NE * labelscalefactor); dot((5.46,0.26),dotstyle); label("$K$", (5.51,0.34), NE * labelscalefactor); dot((2.9,2.78),dotstyle); label("$L$", (2.95,2.87), NE * labelscalefactor); dot((4.81,0.9),dotstyle); label("$O_1$", (4.86,0.97), NE * labelscalefactor); dot((3.75,1.95),dotstyle); label("$O_2$", (3.81,2.03), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* re-scale y/x */ currentpicture = yscale(1.03) * currentpicture; /* end of picture */[/asy][/asy]