Circle geo (1 Viewer)

[terimaa]

Hey guys,
to prove something is a cyclic quad is it sufficient to show there is angle in the semi circle

E.g hsc 4u 2009; part ii
Angle acb= 90 (angle in a semi circle is 90)
Angle kcx= 90(angle in a straight line are supp)
Similarly kdx = 90

Therefore ckdx is a cyclic quad with diameter kx

Cheers guys!!!!

 

[Sy123]

Hey guys,
to prove something is a cyclic quad is it sufficient to show there is angle in the semi circle

E.g hsc 4u 2009; part ii
Angle acb= 90 (angle in a semi circle is 90)
Angle kcx= 90(angle in a straight line are supp)
Similarly kdx = 90

Therefore ckdx is a cyclic quad with diameter kx

Cheers guys!!!!

You have assumed that B, C, K are all collinear (second step) it is not mentioned in the question and is in fact proven in the last part.

To prove it is a cyclic quadrilateral, first we notice that angle ABD is equal to ACD (angle same segment)
Moreover, angle ACD is equivalent to angle XCD, since X, A and C is collinear.
From part (i) angle ABD is equal to angle XKD, thus angle XKD is equal to XCD
Thus converse of angle in same segment, thus cyclic quadrilateral.