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Translating and analyzing

I’m translating the problem into English, noting the triangle ABC with point J at its tangent circle's center. Points P, B, C, and Q are colinear, and PB equals AB, QC equals AC. The task is to prove the sum of angles BAC and QJP equals 180°.

Mapping the relationship

I’m thinking through triangle ABC with a circle tangent to side BC and extensions of sides AB and AC. Points P, B, C, Q lie in this order on a line, and I’m asked to prove angle BAC + angle QJP = 180°.

Analyzing the configuration

Point J is the excenter opposite vertex A of triangle ABC, tangent to side BC and the extensions of sides AB and AC, while points P, B, C, and Q lie in that order on a straight line.

Piecing together angle relationships

I’m setting out to show that the sum of angles BAC and QJP equals 180 degrees, focusing on excenter and excircle properties of triangle ABC.

Mapping out triangle ABC

I’m working on identifying the excenter opposite vertex A in triangle ABC, where P, B, C, and Q are colinear, with PB equaling AB and QC equaling AC.

Setting up coordinates

OK, let me see. I’m establishing coordinate axes with BC on the x-axis and points B and C placed. Considering excenter at the intersection of the external angle bisectors of angles B and C and internal angle bisector of angle A.

Mapping out geometric concepts

I'm working through angle QJP using triangle ABC's excentral triangle and excenters. Considering its relationship with angle A and its supplementary nature to angle A.

Piecing together properties

I’m gathering insights on angle QJP potentially equaling 180° minus angle BAC, and examining similarities between triangles or excenter properties.

Examining excenter properties

I’m mapping out the excenter opposite vertex A in triangle ABC. It’s where the internal angle bisector of angle A intersects the external angle bisectors of angles B and C.

Calculating excenter coordinates

I’m working through the excenter opposite vertex A by locating it on the external angle bisectors of angles B and C. This involves coordinate geometry with triangle ABC and its side lengths.

Connecting external angle bisectors

I'm exploring the links between the excenter opposite vertex A, the external angle bisectors of angles B and C, and their associated distances in triangle ABC.

Mapping out circle and tangency

I’m thinking through a circle centered at J, tangent to side BC at point D, and tangent to extended sides AB and AC at points E and F. This setup feels crucial for establishing relationships between points and distances.

Charting the path

I’m figuring out the coordinates of points P, B, C, and Q on a line to express the total length PQ as the sum of PB, BC, and CQ. Now, I’m working on identifying the position of point J on this line.

Examining excenter J

I’m exploring the excenter J in triangle ABC, considering its position along external angle bisectors and its distances JB and JC. This brings me to the excentral triangle.

Examining excenter distances

I’m trying to figure out the distance of the excenter opposite vertex A from the sides of the triangle, using the semiperimeter and side lengths as key variables.

Analyzing triangle properties

I’m thinking through the excircle radius rara​ in triangle ABC, which is Areas−as−aArea​, and examining its distances from sides and vertices.
Speculating on excenter and tangency

I’m thinking through the excenter J's location on the y-axis or elsewhere. The circle centered at J is tangent to side BC at D, and tangents extend to points E and F on lines AB and AC respectively.

Piecing together possibilities

I'm noting that the circle centered at J is tangent to side BC at point D, with JD perpendicular to BC. This leads me to explore triangle JBC and the angles between lines QJ, PJ, JQ, and JP.

Connecting angles

I’m thinking about proving the angle between lines JQ and JP equals angle A, and then using that to show angle QJP and angle A add up to 180°.

Analyzing external angle bisectors

I’m examining the excenter J opposite vertex A, noting the external angle bisectors from vertices B and C converge at point J and the angles they form with side BC.

Reassessing angle relationships

Piecing together the angle between JB and JC, I’m thinking through the difference between 180° and half the sum of angles A and C, simplified to 90° plus half of angle A.

Analyzing angle relationships

I'm mapping out the angle between lines JP and JQ, considering the positions of P, B, C, Q, and J, and examining the properties of external angle bisectors to find a solution.

Determining excenter J

I’m piecing together the direction vectors for sides BA and BC in the triangle, attempting to find the external angle bisector of angle B, but the calculations are intricate.

Examining trilinear coordinates

I’m looking into trilinear coordinates since the excenter J opposite A is given by (-a : b : c). Also, I’m considering known theorems or properties to simplify the problem.

Wondering about angles

I'm thinking about the angles around point J in triangle ABC and the excenter opposite vertex A, aiming to show that the angle between lines JP and JQ is angle A.

Imagining the scenario

OK, let me see. The circle centered at J is tangent to BC at D and extends to AB and AC at E and F, Points P and Q.

Piecing together tangency

I'm working through the idea that P and Q are tangency points of the excircle with extensions of sides AB and AC. Lines JP and JQ could then be tangents to the circle centered at J.

Exploring geometric relationships

I’m examining the connection between the angle at J and the internal angle bisectors at B and C, while considering circle tangents from P and Q.

Taking a closer look

I explored how the angle between lines JP and JQ could be 90° plus half of angle A, but found this inconsistent.

Reassessing angles

I’m thinking about the angles JP and JQ, considering different external angle bisectors and various angle relationships, but struggling to connect these ideas effectively.

Analyzing angle relationships

I’m mapping out the angles between JB, JP, JC, and JQ at point J, using α and β to express angles between JP and JQ. The goal is to derive an expression for the angle between JP and JQ.

Weighing the angles

I’m comparing angles B and C to JQ and JP. Simplifying the sum using half-angles, the angle between JP and JQ turns out to be zero degrees, which seems incorrect.

Mapping angles

I'm digging into the angles at point J in triangle ABC, focusing on JB and JC, and how they relate to the angle bisectors. It's worth noting that the angle between JB and JC is 90° - (1/2) angle A.

Reevaluating assumptions

I'm thinking through the assumption about the angle between JB and JC, and it turns out JP and JQ are colinear, contradicting the initial assumption.

Revisiting the angle

I’m rethinking the angle between the external angle bisectors at B and C, concluding it’s 90° - (1/2) angle A, based on the relationship between JB, JC, KA, and CD.

Analyzing angles

Thinking through the angle between JP and JQ, it seems to be zero degrees. This leads me to consider if P, J, and Q are colinear or if there's a calculation issue.

Summing angles

I’m piecing together the angles at point J. By adding θ_B, θ_A, and θ_C, they indeed total 180°, confirming their alignment in a plane.

Revisiting the approach

OK, let me see. The angle between JP and JQ is determined to be 0°, confirming they are colinear. This indicates a need to re-evaluate the strategy.

Reaching a conclusion

I'm concluding that the points P and Q refer to E and F, where the circle is tangent to the extensions of sides AB and AC.