MathDB

Triangle

ABC

is inscribed in a circle with center

O'

. A circle with center

O

is inscribed in triangle

ABC

AO

is drawn, and extended to intersect the larger circle in

D

. Then, we must have:

\text{(A)}\ CD=BD=O'D \qquad \text{(B)}\ AO=CO=OD \qquad \text{(C)}\ CD=CO=BD \qquad\\ \text{(D)}\ CD=OD=BD \qquad \text{(E)}\ O'B=O'C=OD

[asy] size(200); defaultpen(linewidth(0.8)+fontsize(12pt)); pair A=origin,B=(15,0),C=(5,9),O=incenter(A,B,C),Op=circumcenter(A,B,C); path incirc = incircle(A,B,C),circumcirc = circumcircle(A,B,C),line=A--3*O; pair D[]=intersectionpoints(circumcirc,line); draw(A--B--C--A--D[0]^^incirc^^circumcirc); dot(O^^Op,linewidth(4)); label("

A

",A,dir(185)); label("

B

",B,dir(355)); label("

C

",C,dir(95)); label("

D

",D[0],dir(O--D[0])); label("

O

",O,NW); label("

O'

",Op,E);[/asy]

Problem Statement