Question

***

Answer 1

Hint : circumcenter of an acute triangle

Answer 2

Hint2 : the incenter of triangle ABC is same as the circumcenter of triangle DEF

Answer 3

the vertices of the triangle DEF lie on the circle
so lets assume that the triangle DEF is obtuse
any obtuse triangle in a circle wuld be like-

Answer 4

is this drawing appearing properly ?^

Answer 5

so now i have to prove that the center of the circle is inside DEF... but how?

Answer 6

well we know that the circle is tangent to triangle at D,E,F

so the sides AB,BC,CA are tangents to the incircle

the lines AB , BC, CA are represented by L1, L2,L3

but we can easily see that they do not form a triangle that keeps the circle in it ..

Answer 7

sorry... why not?

Answer 8

ok so lets extend the tangents and the point where they meet joing them would give us the triangle -