如图，在平行四边形ABCD中，AC、BD相交于点O，∠AOB=∠ABC。求证：BD^2=2BC^2
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△∵∠ABC=∠AOB,∠BAC=∠OAB,∴△AOB∽△ABC∴BO/BC=AO/AB=AB/AC∵AC=2AO∴AB^2=2AO^2∴AO^2/AB^2=1/2∴BD^2/BC^2=(2BO)^2/BC^2=4AO^2/AB^2=4/2=2即BD^2=2BC^2
如图，在平行四边形ABCD中，AC、BD相交于点O，∠AOB=∠ABC。求证：BD^2=2BC^2
△∵∠ABC=∠AOB,∠BAC=∠OAB,∴△AOB∽△ABC∴BO/BC=AO/AB=AB/AC∵AC=2AO∴AB^2=2AO^2∴AO^2/AB^2=1/2∴BD^2/BC^2=(2BO)^2/BC^2=4AO^2/AB^2=4/2=2即BD^2=2BC^2