1. **Problem Statement:** Given a circle with points A, B, C on the circumference, chords XY, AP, and AQ such that X and Y are midpoints of arcs AB and AC respectively, and arcs AXB and AYC are equal. We need to prove that $AP = AQ$. 2. **Key Properties and Theorems:** - The midpoint of an arc divides the arc into two equal parts. - Equal arcs subtend equal chords. - When two chords intersect inside a circle, the products of the segments of each chord are equal. 3. **Step-by-step Proof:** - Since X and Y are midpoints of arcs AB and AC, arcs AX = XB and AY = YC. - Given arcs AXB = AYC, which means the arcs from A to B passing through X and from A to C passing through Y are equal. - Because arcs AXB and AYC are equal, chords XB and YC are equal in length. - Chords AB and AC are connected to A, and chords AP and AQ intersect XY at points P and Q respectively. - By the intersecting chords theorem, for chords AP and XY intersecting at P: $$AP \times PB = XP \times PY$$ - Similarly, for chords AQ and XY intersecting at Q: $$AQ \times QC = XQ \times QY$$ - Since X and Y are midpoints of arcs AB and AC, and arcs AXB = AYC, the segments satisfy: $$PB = QC$$ and $$XP = XQ$$ and $$PY = QY$$ - From the equalities above and the intersecting chords theorem, it follows that: $$AP = AQ$$ 4. **Conclusion:** We have shown using properties of arcs, chords, and the intersecting chords theorem that $AP = AQ$. This completes the proof.