Problem 1

Let

A,B,C,D

be four distinct points on a line, in that order. The circles with diameters

AC

and

BD

intersect at

X

and

Y

. The line

XY

meets

BC

Z

. Let

P

be a point on the line

XY

other than

Z

. The line

CP

intersects the circle with diameter

AC

C

and

M

, and the line

BP

intersects the circle with diameter

BD

B

and

N

. Prove that the lines

AM

DN

, and

XY

are concurrent.

Step 3 of 3: Use the radical-axis power identity

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In plain words

The two circles have common chord $XY$ , so every secant through $Z$ has the same power.

BZ\cdot DZ=XZ\cdot YZ=AZ\cdot CZ

Detailed analysis

Power of $Z$ with respect to the two circles gives $BZ\cdot DZ=XZ\cdot YZ=AZ\cdot CZ$ . Consequently $QZ=Q'Z$ , so $Q=Q'$ . This common point lies on $DN$ , $AM$ , and $XY$ , proving concurrency.