Circle Geometry: Trivial Question on Chords? (1 Viewer)

[Lukybear]

Two chords AB and CD bisect each other at X. Show that this is not possible unless X is the center.

NO idea how to approach this one as well.

Also

Show that in a circle, a shorter chord is always further from the centre than a longer chord.

 

[hscishard]

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Suppose x is not the centre

AX = BX
CX = DX
Angle CXB = AXD (opp ang)

AXD congruent to BXC

Let angle DCB = y
Then angle CDA = y (corresp ang in congruent triangles)
Angle CBA = y (Same segment)
Angle BAD = y (Same segment [to ang BCD]

Then AX =XC, BX = XD (isosceles)
Therefore AX = XC = BX =XD ( corresp sides in congruent triangles)

This only happens when they are radii, hence X coincides with O, the centre.
No clue how to prove the radius bit.