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For family of circles whose inscribed angle makes an angle of $ \theta$ with a line segment is given as: enter image description here If the coordinate of $E=(x_1,y_1)$ and of $B=(x_2,y_2)$ then the family of circles for which $EBH$ makes an angle $\theta$ (here 53) is given as: $$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) \pm \cot \theta \left[ (x-x_1)(y-y_2) - (x-x_2)(y-y_1)\right]$$

This result was given a JEE book (Famous Indian engineering exam) with no explanation why this is true, could any one shed light on why this formula represents what it is supposed to?

tryst with freedom
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  • @Pravimish all points standing on the same arc make the same angle for a given circle of radius, look up inscribed angle theorem. Else, explain why you think it's wrong – tryst with freedom Feb 06 '21 at 21:07
  • sry for the confusion, ihave deleted the comment – Pravimish Feb 07 '21 at 09:01
  • Also, can u give some clarity on the following thought. U have a fixed line segment EB. If this always forms an angle of $53^{\circ}$, then isnt there only one possible triangle, ergo, one possible circle. – Pravimish Feb 07 '21 at 09:05
  • take a point further outside, that would make a different angle while standing on the same arc, no? Now apply the inscribed angle thinking to that point as well @Pravimish – tryst with freedom Feb 07 '21 at 10:09

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