MathLabs

Problem 1

Let A,B,C,DA,B,C,D be four distinct points on a line, in that order. The circles with diameters ACAC and BDBD intersect at XX and YY. The line XYXY meets BCBC at ZZ. Let PP be a point on the line XYXY other than ZZ. The line CPCP intersects the circle with diameter ACAC at CC and MM, and the line BPBP intersects the circle with diameter BDBD at BB and NN. Prove that the lines AMAM, DNDN, and XYXY are concurrent.
Step 2 of 3: Locate the intersection from AMAM
Previous stepNext step
In plain words

The same similarity argument on the other diameter circle gives the analogous product.

Q′Z=AZ⋅CZPZQ'Z=\frac{AZ\cdot CZ}{PZ}
Detailed analysis

Let Q′=AM∩XYQ'=AM\cap XY. Repeating the preceding angle and similarity argument with A,C,MA,C,M gives Q′Z=AZ⋅CZPZQ'Z=\frac{AZ\cdot CZ}{PZ}.