Question

If the diagonal BD of a quadrilateral ABCD bisects both angle B and angle D, show that
$\frac{ab}{bc} = \frac{ad}{cd}$
( please ignore that the line segments are not named in CAPITALS )

Best answer will be marked as the Brainliest answer..

Answer 1

let ABCD be a quadrilateral wherein AB=AD and CB=CD.
To prove : AC is the perpendicular bisector of BD Consider triangle ADB , as AB=AD, it is an isosceles triangle.
=> by property of isosceles triangle , angle ADB = angle ABD therefore, triangle ADB is similar to triangle ABD. Now of similar triangle => side OD = side OB.
=> AB/OB = AD/OD
=> AO is a bisector of BD. similarly, in triangle BCD, BC= CD
=> it is also isosceles triangle , therefore angle CDB=angle CBD.and hence , triangle CDB is similar to triangle CBD
=> side OB= side OD.
=> CD/OD = CB/OB
=> CO is a bisector of BD. As OA and OC is the bisector of triangle ABD and triangle BCD respectively.
Therfore AC is a bisector of BD and perpendicular to BD.
Hence proved.