8. I prove that, the two intersecting chords of any circle can not bisect each other unless both of them are diameters of the circle
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Let AB and CD be two chords intersecting at point O
In △AOC and △DOB
1) =∠AOC=∠DOB (vertically opposite angles)
2) OD=OC Radins of circle
3) AO=OB Radins of circle
∴ △AOC≅△DOB (SAS congruency)
∴
AC
^
=
BD
^
−−−(1)
In △AOD and △BOC,
proceeding similarly △AOD≅△BOC
∴
AD
^
=
BC
^
−−−−(2)
(1)+(2),
AC
^
+
AD
^
=
BD
^
+
BC
^
or,
CAD
^
=
CBD
^
i.e., CD divide circle into 2 semi-circles,
∴ CD is a diameters
Similarly, AB is a diameters.
Proved.
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