Question 6543: Show that quadrilateral ABCD can be inscribed in a circle if and only if angle B and angle D are supplementary.
Hint: To proveif, show that D lies on the unique circle through A, B, and C.
Note: A quadrilateral inscribed in a cricle is said to be cyclic.
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Here I you the sketch of the proof:
Show that quadrilateral ABCD can be inscribed in a circle if and only if angle B and angle D are supplementary.
--> If ABCD can be inscribed in a circle , then
angle ABC = 1/2 arc BDC and angle ADC = 1/2 arc ABC.
But, arc BDC + arc ABC = whole circle = 360 deg
Hence, angle B + angle D = 180 deg
<-- Draw a circle pass through A,B & C.
If point D is lying inside the circle,then
angle D + angle B > 180 (why?, hint: by extension line AD to
intersect the circle at point E)
If point D is lying outside the circle,then
angle D + angle B < 180 (why?)
Hence, we conclude that D must be lying on the circle.
Kenny
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