In ΔABC, X ∈ BC and B-X-C. A line passing through X and parallel to AB intersects AC in Y. A line passing through X and parallel to BY intersects AC in Z. Prove that CY²=AC.CZ.
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Given : in ∆ABC, X ∈ BC and B-X-C. A line passing through X and parallel to AB intersects AC in Y. A line passing through X and parallel to BY intersects AC in Z.
To prove : CY² = AC.CZ
proof : in ∆ABC, XY || AB
according to Thales theorem,
AY/CY = BX/CX
(AY + CY) = (BX + CX)/CX
AC/CY = BC/CX
CY/AC = CX/BC ................(i)
in ∆ABC, XZ || BY
according to Thales theorem,
YZ/ZC = BX/CX
(YZ + ZC)/ZC = (BX + CX)/CX
CY/ZC = BC/CX
ZC/CY = BX/BC ..........(ii)
from eqs. (i) and (ii),
CY/AC = ZC/CY
CY² = AC. CY
hence proved
To prove : CY² = AC.CZ
proof : in ∆ABC, XY || AB
according to Thales theorem,
AY/CY = BX/CX
(AY + CY) = (BX + CX)/CX
AC/CY = BC/CX
CY/AC = CX/BC ................(i)
in ∆ABC, XZ || BY
according to Thales theorem,
YZ/ZC = BX/CX
(YZ + ZC)/ZC = (BX + CX)/CX
CY/ZC = BC/CX
ZC/CY = BX/BC ..........(ii)
from eqs. (i) and (ii),
CY/AC = ZC/CY
CY² = AC. CY
hence proved
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