October 2026 Problem of the Month

The diagonals of a square meet at O.

The bisector of angle OAB meets BO and BC at N and P respectively.

The length of NO is 24 cm.

How long is PC?

Solution to the Problem:

Here is the answer:

PC is 48 cm long.

Draw a line segment horizontally to the left from O, meeting line AP at point Z.



Triangle ZON is isosceles, because its angles are 67.5, 45, 67.5 degrees. So the length of ZO equals the length of NO, which is 24 cm.

Triangles AZO and APC are similar, and the length of AC is twice the length of AO, so the length of PC is twice the length of ZO.

That means the length of PC is 48 cm.

Originally, I thought the problem needed trigonometry to be solved without drawing a segment...

But Dr. Kishan proved me wrong -- twice.   The problem CAN be solved without having to draw an auxillary line and without having to use trig.
Click on the links below to see Dr. Kishan's two solutions to the problem.
Click here to see solution #1
Click here to see solution #2