In the triangle ABC, the angle c = 90 ° and the angle B = 45 ° ad bisects the angle BAC and intersects BC at the point D. It is proved that ab = AC + CD

In the triangle ABC, the angle c = 90 ° and the angle B = 45 ° ad bisects the angle BAC and intersects BC at the point D. It is proved that ab = AC + CD

The CAD is 1 and the DAE is 2
Intercept AE = AC, connect De
∵ ad bisection ∠ BAC,
∴∠1=∠2.
In △ ACD and △ AED, AC = AE, ﹥ 1 = ﹥ 2, ad = ad, ﹥ ACD ≌ AED (SAS). ﹥ AED = ﹥ C = 90, CD = ed,
And ∵ AC = BC, ∵ B = 45 °
∴∠EDB=∠B=45°.
∴DE=BE,
∴CD=BE.
∵AB=AE+BE,
∴AB=AC+CD.