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 ......