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 2 of 5: Circle through A, C, D

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x^2+y^2+(d-c)x-\frac{a^2-cd}{a}y-cd=0

Detailed analysis

Substituting $A$ , $C$ , $D$ into the general circle equation $x^2+y^2+2fx+2gy+h=0$ and solving the resulting linear system for $f,g,h$ gives the circumcircle of $\triangle ACD$ shown above.