Question

In triangle ABC. Angle C is an acute angle and angle C = 2angle B.
AD is perpendicular to BC

Prove that AC = BD - DC.

Answer 1

In the right angle triangle ABD, we have

AD² = AB² - BD²

In the right angle triangle ACD we have

AD² = AC² - CD²

So AC² - CD² = AB² - BD²

=> AC² = AB² + CD² - BD²

=> = AB² + (CD + BD)* (CD - BD )

= AB² + BC * [(BC - BD) - BD]

= AB² + BC * [ BC - 2 BD ]

= AB² +BC² - 2 BC * BD