Thread: 3 Circle Geometry Proofs

Hi, here are 3 sums which I need help in proving.

1) PQ is a tangent to the circle at A and O is the center of the circle. If BC is parallel to PQ, prove that angle BOA = 2x angle CAQ.


2) AC is a tangent at A to the circle, center O. BC is perpendicular to CA. Prove that BC is parallel to OA and AB bisects angle OBC.


3) The chords AB and CD intersect at E and the tangent PA is parallel to chord CD. Prove that AB bisects angle CBD.


Thanks for your explanations.

Hello

1) AO is an axis of symmetry of the figure => angle BOA=angle COA
OC=OA => triangle COA is isosceles => angle OCA=angle OAC

Sum of angles of triangle COA=180°
angle COA + angle OCA + angle OAC = 180°
angle COA + 2 x angle OAC = 180° (1)

PQ is tangent to the circle => angle OAQ = 90°
But angle OAQ = angle OAC + angle CAQ = 90°
Therefore angle OAC = 90° - angle CAQ (2)

(1) and (2)
angle COA + 2 x (90° - angle CAQ) = 180°
angle COA - 2 x angle CAQ = 0