A mathematical problem: let p be a point inside the square ABCD, and the distances from P to vertex a, B and C are 1, 2 and 3, respectively

A mathematical problem: let p be a point inside the square ABCD, and the distances from P to vertex a, B and C are 1, 2 and 3, respectively

Rotate △ APB around point B to make AB coincide with BC and P coincide with point Q. connect PQ
It is easy to prove that △ PBQ is isosceles right triangle,
PQ = 2 root 2
According to the inverse theorem of Pythagorean theorem, it is obtained that PQC = 90 °
∴∠APB＝∠BQC＝135°
Cross point a as am ⊥ BP, and extend the long line to point M,
Then △ APM is an isosceles right triangle,
So, AP = PM = root 2 / 2
Ψ BM = 2 + radical 2 / 2
In △ ABM, according to Pythagorean theorem
AB = root (am ＾ 2 + BM ＾ 2) = under root (5 + 2 √ 2)