a circle inscribed in quadrilateral ABCD at points P,Q,R,S then prove that AB+CD=BC+DA
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Given: ABCD is a quadrilateral
To prove: AB+CD=BC+DA
Proof:-
AP=AS (length of tangents are equal) ___1
BP=BQ ( '' '' '' '' )____2
CR=CQ('' '' '' '' )____3
DR=DS('' '' '' '' )____4
eq 1+2+3+4 =>
AP+BP+CR+DR=AS+BQ+CQ+DS
(AP+BP)+(CR+DR)=(AS+DS)+(BQ+CQ)
AB+CD=AP+BC
=> AB+CD=BC+DA
To prove: AB+CD=BC+DA
Proof:-
AP=AS (length of tangents are equal) ___1
BP=BQ ( '' '' '' '' )____2
CR=CQ('' '' '' '' )____3
DR=DS('' '' '' '' )____4
eq 1+2+3+4 =>
AP+BP+CR+DR=AS+BQ+CQ+DS
(AP+BP)+(CR+DR)=(AS+DS)+(BQ+CQ)
AB+CD=AP+BC
=> AB+CD=BC+DA
taqia:
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