Mathematics

Problem

Consider

\triangle ABC

with

\angle B = 90^\circ

. The perpendicular from

B

to the hypotenuse

AC

meets

AC

D

. a. (i) Prove that

\triangle ADB \sim \triangle ABC

and

\triangle CDB \sim \triangle ABC

. b. (ii) Using the similarities established in part (i), derive the relations

AD \cdot AC = AB^2

and

DC \cdot AC = BC^2

. c. (iii) From the relations obtained in part (ii), deduce that

AC^2 = AB^2 + BC^2