Answer:
Step-by-step explanation: He has to draw a perpendicular line (dot line in Annex) from A vertex to side BC. Doing that, he will build two additional triangles ADB and ADC.
We can show these three triangles are similar.
In triangle ABC angles
∠ BAC is a right angle (90° )
Then we can call ∠ ACB = α and therefore ∠ ABC = 90 - α
Since the sum of internal angles in any triangle is 90°
Now in Triangle ADB
∠ ADB is a right angle (90° )
Sides of ∠ BAD are perpendicular sides of ∠ ABC according to
AB ⊥ AC and BC ⊥ AD Then ∠ BAD = ∠ ABC = 90° - α
And triangle ADC
∠ ADC is a right angle (90° )
∠ ACB is common to both triangles ( the original one and triangle ACD )
Then we have shown that the three triangles are similar
Δ ABC ≈ Δ ADC ≈ Δ ADB
Therefore we can work with sides proportionality:
b /a = a - x / b or b² = a ( a - x ) b² = a² - ax (1)
And
c / x = a /c ⇒ c² = ax (2)
Then adding (1) and (2)
b² + c² = a²
What we want to show