Direct Proof

Question

 

A right-angled triangle has sides of length a, b and c, with c being the hypotenuse. If the

triangle has area c2/4, show that it is isosceles.

Answer 1

(1) Pythagoras: a²+b²=c²

(2) area=ab/2=c²/4, b=c²/(2a), substitute for b in (1):

a²+c⁴/4a²=c², 4a⁴-4a²c²+c⁴=0=(2a²-c²)², therefore a²=c²/2, a=c/√2.

Thus b=c²/(c√2)=c/√2=a. a=b so the triangle is isosceles QED