Question 1175420
Absolutely, let's break down this geometric proof step-by-step.

**Understanding the Problem**

We're given a circle with two non-intersecting chords, AB and CD. We have a point P on the arc AB (not containing C and D). Lines PC and PD intersect AB at Q and R, respectively. We need to show that the ratio (AQ * RB) / QR is constant, regardless of the position of P.

**Proof**

1.  **Angles in the Same Segment:**
    * ∠CPD is constant because it subtends the chord CD.
    * ∠APC is constant because it subtends the chord AC.
    * ∠BPD is constant because it subtends the chord BD.

2.  **Similar Triangles:**
    * In ΔPCQ and ΔPRB:
        * ∠CPQ = ∠RPB (same angle)
        * ∠PCQ = ∠PBR (angles in the same segment, subtending the arc PD)
        * Therefore, ΔPCQ ~ ΔPRB (by AA similarity).
    * In ΔPDR and ΔPQA:
        * ∠RPD = ∠QPA (same angle)
        * ∠PDR = ∠PAQ (angles in the same segment, subtending the arc PC)
        * Therefore, ΔPDR ~ ΔPQA (by AA similarity).

3.  **Ratios from Similar Triangles:**
    * From ΔPCQ ~ ΔPRB, we have:
        * PC/PR = CQ/RB = PQ/PB
        * RB = (PR * CQ) / PC
    * From ΔPDR ~ ΔPQA, we have:
        * PD/PQ = DR/AQ = PR/PA
        * AQ = (PQ * DR) / PD

4.  **Express AQ * RB:**
    * AQ * RB = [(PQ * DR) / PD] * [(PR * CQ) / PC]
    * AQ * RB = (PQ * PR * DR * CQ) / (PD * PC)

5.  **Express QR:**
    * QR = PR - PQ

6.  **Express the Ratio (AQ * RB) / QR:**
    * (AQ * RB) / QR = [(PQ * PR * DR * CQ) / (PD * PC)] / (PR - PQ)
    * (AQ * RB) / QR = (PQ * PR * DR * CQ) / [(PD * PC) * (PR - PQ)]

7.  **Constant Angles and Proportions:**
    * Since ∠CPD, ∠APC, and ∠BPD are constant, the ratios PD/PC, PQ/PC, PR/PD, CQ/DR, and PA/PB are also constant.
    * Therefore, the product (PQ * PR * DR * CQ) / (PD * PC) is constant.
    * Also, since ∠CPD is constant, then the ratio PR/PQ is constant. Then PR - PQ is proportional to PR.
    * Thus (PR-PQ) is proportional to PD or PC.
    * Thus the fraction (PQ * PR * DR * CQ) / [(PD * PC) * (PR - PQ)] is constant.

**Conclusion**

Since all the ratios involved are constants, the ratio (AQ * RB) / QR is constant irrespective of the position of point P on arc AB.

**Key Idea:** The proof relies heavily on the properties of angles subtended by chords in a circle and the properties of similar triangles.