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 3 of 5: Circle through B, D, E

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x^2+y^2+dx-\frac{a^2-d^2}{2a}y=0

Detailed analysis

The same method with $B$ , $D$ , $E$ in place of $A$ , $C$ , $D$ gives the circumcircle of $\triangle BDE$ shown above; it passes through the origin, so its constant term is $0$ .