Answer :

i) We know that,


In any triangle, line drawn parallel to one side, passing through mid-point of another side will also meet the third side at its mid-point.



Therefore,


Let AD be ‘x’, then DC = x. similarly AE = y then EB = y


By pythagorus theorem for bigger triangle ABC,




----1


Similarly for smaller triangle AED,




-----2


From 1 & 2, we have



Hence proved.


ii)


We have the above following data.


And from pythagorous theorem,


We have






Hence the distance between mid-point of hypotenuse to vertex B (or) A (or) C is x


Hence proved.


iii) We know that


circumcentre is a point from which all the vertices are at equal distances.


In the above case, we have proved the mid-point of hypotenuse is equidistant from all the three vertices. Therefore mid-point of hypotenuse is the circumcentre of the given triangle.



Hence proved.


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