the bisectors of the opposite angles A and C of a cyclic quad intersect at E and F.prove that EF is diameter?

Question

this is a question from ncert maths textbook

Answer 1

opposite angles of a cyclic quadrilateral are supplementary, so ang.A + angC=180
ang A= 180 - ang C
ang C= 180 - ang A
IN TRIANGLE DAC
ang DAC= (180 - ang C)/2
ang DCA=(180 - ang A)/2
ang D = 180-(180 - ang C)/2 + (180 - ang A)/2
= 180-(180-(ang C + ang A))/2
= 180-(180-90)
= 90
AS ang d = 90 , only diameter subtends ang 90 at any point on circle, so ef(or ac) is diameter

Answer 2

It looks like quadrilateral ABCD is inscribed in a circle. You probably have drawn a circle with the bisectors properly labeled. Then the problem is easy (although explaining it is tough!) Since angle DAE = angle EAB (bisected) arc DE = arcECB. By the same reasoning arc FB = arc FAD. Therefore, arc FAD + arc DE = arc FB + arc ECB. Do a little substituting and use whatever theorem you have for proving that a chord is a diameter if it subtends two equal arcs. Good problem!

Answer 3

Let G be the center of the circumscribing circle.
Then angle DGF = arc FD and angle DGE = arc DE
Angle DCF = 1/2 arc FD and angle DAE = 1/2 arc DE
Therefore angle DGF+angle DGE = 2(angle DCF+angle DAE)
But 2(angle DCF+angle DAE) = angle DCB+angle DAB = 180 degrees because opposite angles in inscribed circle are supplementary.
Therefore angle DGF+angle DGE = 180 degrees
Therefore angle FGE = angle DGF+angle DGE is a straight angle.
Therefore EF is a diameter because it is a straight line passing through the center of a circle (definition of diameter.).

Answer 4

quadrilateral ABCD is inscribed in a circle.
angle DAE = angle EAB (bisected) arc DE = arcECB. By the same reasoning arc FB = arc FAD. Therefore, arc FAD + arc DE = arc FB + arc ECB. use whatever theorem you have for proving that a chord is a diameter if it subtends two equal arcs.
interesting!

Answer 5

in a cyclic quadrilateral sum opp pairs of angles are 180.
hence a+c=180
1\2a+1\2c=90
hence in a cir a straight line {180} and hence it is the longest straight that can be drawn in a circleand therefore a longest line must pass through the centre it must be diametre