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Quadrilateral ABCD is a cyclic quadrilateral, 

Prove that: (AB ∗ CD) +(BC ∗AD) = (AC ∗ BD)

james bond

Hi james bond !
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pls solve this

Mukesh Jha     view comments

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Hi james bond !
Mark as answer and close

 

 

Given: Quadrilateral ABCD is a cyclic quadrilateral.

To prove: (AB * CD)+(BC * AD) = (AC * BD)

Proof: 

Quadrilateral ABCD is a cyclic quadrilateral.......... (Given)

Extent BA to M such that 

LBDM = LCDA................... (1) (By construction)

mLDCB + mLDAB = 1800..................(2) (Sum of the opposite angles of a cyclic quadrilateral is supplementary)

mLDAM + mLDAB = 1800..................(3) (Sum of the opposite angles of a cyclic quadrilateral is supplementary)

Therefore, LDAM = LDCB .................................(4) (From 1, 2 and 3)

Therefore, triangle DCB = triangle DAM................... (by AA test)

Therefore, DC/DA = CD/AM .........................(by Corresponding sides of similar triangle)

Therefore, AM = [(CD x DA)/DC]........................... (5)

LDCA =LDBA...................(6) (angle subtended by same arc)

LCDA...................(7) (from 1)

Traingle CDA similar Triangle BDM.................. (8)(By AA test)

DC/DB = CA/BM

BM = [(CA x DB)/ DC]..........................(9)

BM = BA +AM .............( By Construction)

[CA x DB] / DC = BA + (CD x DA)/DC

(CA x BD) = (BA x DC) + (CD x DA)

 

 

 

 

ULA Admin     view comments

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