Problem 2

Let

ABC

be a right triangle with

\angle B=90^\circ

. Point

D

lies on the line

CB

such that

B

is between

D

and

C

. Let

E

be the midpoint of

AD

and let

F

be the second intersection point of the circumcircle of

\triangle ACD

and the circumcircle of

\triangle BDE

. Prove that as

D

varies, the line

EF

passes through a fixed point.

Step 1 of 5: Set up coordinates

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B=(0,0),\ A=(0,a),\ C=(c,0),\ D=(-d,0),\ E=\left(-\tfrac{d}{2},\tfrac{a}{2}\right),\quad a,c,d>0

Detailed analysis

Place $B$ at the origin with $BA$ along the positive $y$ -axis and $BC$ along the positive $x$ -axis, so $A=(0,a)$ , $C=(c,0)$ . Since $B$ lies between $D$ and $C$ , $D=(-d,0)$ for some $d>0$ . Then $E$ , the midpoint of $AD$ , is $\left(-\tfrac{d}{2},\tfrac{a}{2}\right)$ .