Proving Tangency and Angle Congruence in Triangle Circles?

[app_pp]

Homework Statement

Bisectors of angles A and C of triangle ABC intersect the circumcircle
of this triangle at points P and Q, respectively. The straight
line passing through the incentre (I) of ABC and parallel to AC intersects
the line PQ at point Z. Prove that a) the angle BZQ =QZI
b) the line ZB is tangent to the circumcircle

Homework Equations

Not many equations, maybe the chord theorems, and the congruence tests.

The Attempt at a Solution

Well for 'a' we just need to prove if BI is perpendicular to PQ
we know ZQA is equal to c/2

 

[LightHero]

, and ZQI is equal to a/2,so BQI is equal to (c-a)/2now let's prove thatBI is perpendicular to PQWe know that BP=PI (chord theorem)Now, since PQ is equal to c, we can sayPB=PQ-PI=>PB=c-PI=>PB=c-BQI (since BP=PI)=> PB= c-(c-a)/2=>PB= (c+a)/2Now, let's draw the line PQ and BI intersecting at BWe know that,the angle PBQ = angle BQI =>(c+a)/2 = (c-a)/2=>c = awhich is true, so BI is perpendicular to PQ,So, angle BZQ = angle QZIFor 'b', we need to prove that ZB is tangent to the circumcircle.First, we need to prove that ZI is perpendicular to PQwe know that ABA' = 180=>A + B' = 180=>A + A' = 180=> 2A = 180=> A = 90=> ZI is perpendicular to PQNow, let's draw a line perpendicular to PQ at point Z, which intersects the circumcircle at point BWe know that, ZI is perpendicular to PQ and ZB is perpendicular to PQ=>ZB is tangent to the circumcircleHence, Proved.

 

[Architeuthis Dux]

and since QA is also c/2 (as it is a tangent)
we can get the angle ZQB= c.
Similarly we can prove that ZPB=b and thus the angle BZQ= QZI.

For 'b', since we have already proved that ZBQ and ZBI are congruent, we can use the congruence test for right triangles (Hypotenuse-Leg) to prove that ZB is tangent to the circumcircle. This is because we can show that the length of ZB is equal to the radius of the circumcircle, which is the distance from the center of the circle to any point on its circumference. Therefore, ZB is tangent to the circumcircle at point B.