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If the points (a ,0),(b ,0),(0, c)a n d(...

If the points `(a ,0),(b ,0),(0, c)a n d(0, d)` are concyclic `(a , b , c , d >0)` , then prove that `a b=c ddot`

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Without loss of generality, assume that `bgtaanddgtc`. SinceABCD is cyclic quadrilateral, sum of oppsite angles is `180^@`.

i.e., `angle BAC+ angle CDB=180^@`
Let ` angleBAC=theta` (=inclination of line AC)
`therefore angle CDB=180^@-theta`
Therefore, inclination of line BD is `270^@-theta` slope of line AC.
`tantheta=-(c)/(a)`
Slope of `BD, tan (270^@-theta)=-(d)/(b)`
or `cot theta=-(d)/(b)`
From (1) and (2), we get
`(tantheta)(cottheta)=(-(c)/(a))(-(d)/(b))`
or `ab=cd`
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