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Two Circles Intersect Each Other at Points a and B. a Straight Line Paq Cuts the Circles at P and Q. If the Tangents at P and Q Intersect at Point T; Show that the Points P, B, Q and T Are Concyclic - ICSE Class 10 - Mathematics

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ConceptTangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers

Question

Two circles intersect each other at points A and B. A straight line PAQ cuts the circles at P and Q. If the tangents at P and Q intersect at point T; show that the points P, B, Q and T are concyclic

 

Solution

Join AB, PB and BQ
TP is the tangent and PA is a chord
∴ `∠`TPA = `∠`ABP …… (i) (angles in alternate segment)
Similarly,
`∠`TQA = `∠`ABQ ……. (ii)
Adding (i) and (ii)
`∠`TPA+ `∠`TQA = `∠`ABP + `∠`ABQ
But, Δ PTQ,
`∠`TPA  + `∠`TQA + `∠`PTQ  =180°
⇒ `∠` PBQ  = 180° -  `∠`PTQ
⇒  `∠`PBQ  + `∠`PTQ =  180° ...
But they are the opposite angles of the quadrilateral
Therefore, PBQT are cyclic.
Hence, P, B, Q and T are concyclic.

 

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Solution Two Circles Intersect Each Other at Points a and B. a Straight Line Paq Cuts the Circles at P and Q. If the Tangents at P and Q Intersect at Point T; Show that the Points P, B, Q and T Are Concyclic Concept: Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers.
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