the complete Circle Workbook covering Sections 11–17, based on the interactive HTML resources you provided. All content derives from the following source files:

• tangent-circles-sections11-15-v4.html
• tangent-circles-sections11-16-v4.html
• tangent-circles-sections11-17-v4.html

Each section is described for TES listing use.

SECTION 11 — Tangents (Harder)
Students practise deriving tangent equations to circles in a variety of configurations.
Topics include:
• Tangent to x² + y² = r² at point P(x₁, y₁) using x·x₁ + y·y₁ = r².
• Tangents to circles with shifted centres (x – h)² + (y – k)² = r².
• Equilateral triangle cases via radius and symmetry relationships.
• 10 auto‑marked exercises with diagram prompts.
Skills:
• Tangent-line derivations
• Perpendicular-radius concept
• Substitution into circle equations
• Working with shifted centres

SECTION 12 — Circles through Polygon Vertices
Students identify the circle x² + y² = R² passing through vertices of polygons centred at the origin.
Polygons covered: equilateral triangle, square (side or diagonal known), rectangle, right triangles, isosceles right triangles, and regular hexagon.
Skills:
• Determining radius from geometric structure
• Using geometry → algebra conversion
• Fractional reasoning (coefficients p/q)

SECTION 13 — Incircle of Triangle Formed by Axes and ax + by = c
The triangle formed by the x‑axis, y‑axis and the line ax + by = c has an incircle with centre (r, r), where
r = c / (a + b + sqrt(a² + b²)).
Equation of the incircle:
x² + y² − 2r x − 2r y + r² = 0.
Skills:
• Distance-to-line formula
• Incentre derivation
• Equation of a circle in general form

SECTION 14 — Incircles of General Triangles
Given triangle ABC, students compute:
• Incenter: (a A + b B + c C) / (a + b + c)
• Inradius: r = 2Δ / (a + b + c)
• Circle equation: x² + y² + 2g x + 2f y + c = 0
Skills:
• Coordinate geometry
• Weighted averages (barycentric coordinates)
• Area and inradius computation

SECTION 15 — From Incentre to Circumcentre (Equilateral Case)
If the incircle is x² + y² + 2g x + 2f y + c = 0, then r² = g² + f² – c and the circumradius is R = 2r.
Circumcircle:
x² + y² + 2g x + 2f y + (g² + f² – 4r²) = 0.
Skills:
• Relationship between incircle and circumcircle of equilateral triangles
• Manipulation of general circle equations

SECTION 16 — Inscribed Angles (Angle QPR from Chord QR)
For points Q and R on x² + y² = R², the inscribed angle is:
angle QPR = 1/2 * arccos((Q · R)/R²).
Skills:
• Dot product geometry
• Central vs. inscribed angles
• Chord‑angle relationships

SECTION 17 — Opposite Points & Loci
Part A: Tangents to the circumcircle of a rectangle parallel to its diagonal.
Students compute tangent lines Ax + By ± C = 0.
Part B: A variable circle through P(xP, yP) touching the x‑axis leads to the locus:
(x − xP)² = 4 yP y.
Skills:
• Tangent conditions via distance formula
• Loci from geometric constraints
• Parabolas from circle movement