Hello,

There are assumptions on the orientation of the figure, because there are situations where it doesn't work.

Okay, look at the sketch. M is the perpendicular to BC from P.

CMPN is a rectangle. So in order to prove that PN=AC+CB, it is sufficient and necessary to prove that BM=AC.

And this is easy if you consider triangles BMP and ABC.

- They both have right a right angle.
- Angle ABP is 90° since ABPQ is a square. Hence .

Therefore, the triangles BMP and ABC are similar.

But we know that measures BA and BP are equal since ABPQ is a square. Thus triangles BMP and ABC are congruent.

And we can conclude : BM=AC.

---------> PN=MC=BC+BM=BC+AC ......