Question

In the figure, chord AC and chord BD intersect each other in the point E. Chord AB = chord AD. Prove that AB^2=AE X AC​

Answer 1

Answer:

Quadrilateral ABCD is cyclic......... (given)

therefore by the theorem of cyclic quadrilateral

angle ABC+angleADC=180

therefore angle ABC is 90degree

IN ∆ ABC angle ABC is 90 degree

therefore ∆ABC ~∆AEB~∆BEC .... ( similarity and the theorem of equal ratios)

∆ ABC~∆AEB

therefore

AB/AE=AC/AB=BC/EB ...... (c.s.s.t)

AB/AE=AC/AB

THEREFORE

BY CROSS MULTIPLYING AB^2=AE×AC

HENCE PROVED

Answer 2

Answer:

Step-by-step explanation:

Quadrilateral ABCD IS CYCLIC. ----------------( given )

∴ BY THEOREM OF CYCLIC QUADRILATERAL.

∠ABC + ∠ADC = 180° .

∴∠ABC = 90° .

IN ΔABC , ∠= 90° .

∴ ΔABC ≅ ΔAEB ≅ ΔBEC ---------------- ( SIMILARITY AND THEOREM ON                                                                        EQUAL RATIO )

∴ΔABC ≅ ΔAEB .

∴AB/ AE = AC/AB = BC/EB ------------------- ( C.S.S.T )

∴ BY CROS MULTIPLYING

AB² = AE × AC .

∴ HENCE, PROVED.