MathDB

In the triangle

ABC

\angle A

is biggest. On the circumcircle of

\triangle ABC

, let

D

be the midpoint of

\widehat{ABC}

and

E

be the midpoint of

\widehat{ACB}

. The circle

c_1

passes through

A,B

and is tangent to

AC

A

, the circle

c_2

passes through

A,E

and is tangent

AD

A

c_1

and

c_2

intersect at

A

and

P

. Prove that

AP

bisects

\angle BAC

.
[hide="Diagram"][asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(14.4cm); 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 = -5.23, xmax = 9.18, ymin = -2.97, ymax = 4.82; /* image dimensions */ /* draw figures */ draw(circle((-1.32,1.36), 2.98)); draw(circle((3.56,1.53), 3.18)); draw((0.92,3.31)--(-2.72,-1.27)); draw(circle((0.08,0.25), 3.18)); draw((-2.72,-1.27)--(3.13,-0.65)); draw((3.13,-0.65)--(0.92,3.31)); draw((0.92,3.31)--(2.71,-1.54)); draw((-2.41,-1.74)--(0.92,3.31)); draw((0.92,3.31)--(1.05,-0.43)); /* dots and labels */ dot((-1.32,1.36),dotstyle); dot((0.92,3.31),dotstyle); label("

A

", (0.81,3.72), NE * labelscalefactor); label("

c_1

", (-2.81,3.53), NE * labelscalefactor); dot((3.56,1.53),dotstyle); label("

c_2

", (3.43,3.98), NE * labelscalefactor); dot((1.05,-0.43),dotstyle); label("

P

", (0.5,-0.43), NE * labelscalefactor); dot((-2.72,-1.27),dotstyle); label("

B

", (-3.02,-1.57), NE * labelscalefactor); dot((2.71,-1.54),dotstyle); label("

E

", (2.71,-1.86), NE * labelscalefactor); dot((3.13,-0.65),dotstyle); label("

C

", (3.39,-0.9), NE * labelscalefactor); dot((-2.41,-1.74),dotstyle); label("

D

", (-2.78,-2.07), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]

Problem Statement