Circumcenter of a Triangle

Overview

This animation visually demonstrates the construction and properties of the circumcenter across three triangle types: acute (scalene), right, and obtuse. For each case, it shows how the three perpendicular bisectors meet at the circumcenter and a circumscribed circle is drawn through all three vertices — highlighting the key insight that the circumcenter lies inside an acute triangle, on the hypotenuse of a right triangle, and outside an obtuse triangle.

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Phases

#	Phase Name	Duration	Description
1	Acute Triangle Intro	~2s	A scalene (acute) triangle fades in at the center of the screen with labeled vertices A, B, and C. A case label "Acute Triangle" appears in the top area.
2	First Perpendicular Bisector	~2s	The midpoint of side AB is marked, then its perpendicular bisector is drawn with a right-angle marker at the midpoint.
3	Second Perpendicular Bisector	~2s	The midpoint of side BC is marked and its perpendicular bisector is drawn similarly.
4	Third Perpendicular Bisector	~2s	The midpoint of side CA is marked and its perpendicular bisector is drawn. All three lines visibly converge at an interior point.
5	Highlight Circumcenter (Acute)	~2s	The interior intersection point is highlighted with a distinct dot labeled "O". A brief pulse effect draws attention to it.
6	Equal Radii (Acute)	~2s	Dashed segments are drawn from O to each vertex A, B, and C, with tick marks indicating OA = OB = OC.
7	Circumscribed Circle (Acute)	~2s	The circumcircle is drawn centered at O, passing through all three vertices.
8	Transition to Right Triangle	~2s	All construction lines fade out; the cleaned scene holds briefly, then fades out entirely.
9	Right Triangle Intro	~2s	A right triangle fades in with labeled vertices A, B, C and a right-angle marker at vertex C. The case label updates to "Right Triangle".
10	Perpendicular Bisectors (Right)	~3s	All three perpendicular bisectors are drawn in sequence with midpoint markers and right-angle indicators.
11	Highlight Circumcenter (Right)	~2s	The circumcenter O is highlighted exactly at the midpoint of the hypotenuse AB, making it clear it lies on the hypotenuse. Dashed radii OA = OB = OC are shown.
12	Circumscribed Circle (Right)	~2s	The circumcircle is drawn; it visually coincides with the hypotenuse as its diameter, echoing Thales' theorem.
13	Transition to Obtuse Triangle	~2s	Scene fades out cleanly, transitioning to the final case.
14	Obtuse Triangle Intro	~2s	An obtuse triangle fades in with labeled vertices A, B, C and the obtuse angle prominently marked. The case label updates to "Obtuse Triangle".
15	Perpendicular Bisectors (Obtuse)	~3s	All three perpendicular bisectors are drawn in sequence, visibly extending well beyond the triangle's boundaries to meet at an exterior point.
16	Highlight Circumcenter (Obtuse)	~2s	The circumcenter O is highlighted outside the triangle. Dashed radii from O to each vertex confirm OA = OB = OC.
17	Circumscribed Circle (Obtuse)	~2s	The circumcircle is drawn centered at the exterior point O, passing through all three vertices.
18	Outro	~2s	All construction lines fade out, leaving only the clean obtuse triangle, the exterior circumcenter O, and the circumcircle as a final clear view.

Total: ~38s

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Layout

The animation uses a 9:16 portrait format with a single full-width vertical canvas.

┌─────────────────────────┐
│                         │
│       TOP AREA          │
│  (Case Label: "Acute /  │
│   Right / Obtuse        │
│    Triangle")           │
│                         │
├─────────────────────────┤
│                         │
│                         │
│                         │
│      MAIN VISUAL        │
│   (Triangle, bisectors, │
│    circumcenter, circle)│
│                         │
│                         │
│                         │
├─────────────────────────┤
│                         │
│      BOTTOM AREA        │
│  (Circumcenter position │
│       indicator)        │
│                         │
└─────────────────────────┘

Area Descriptions

Area	Content	Notes
Top	Case label indicating the current triangle type ("Acute Triangle", "Right Triangle", "Obtuse Triangle")	Updates between cases via a crossfade
Main	Triangle with labeled vertices, midpoint markers, perpendicular bisectors with right-angle indicators, circumcenter dot and label "O", radii, and circumcircle	Centered; all construction elements animate sequentially within each case
Bottom	Short indicator of circumcenter position ("Inside", "On Hypotenuse", "Outside")	Updates per case to reinforce the key geometric distinction

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Notes

• For the acute triangle case, the circumcenter O should have a clearly visible interior position — avoid triangles where O is too close to an edge.
• For the right triangle case, the circumcenter coincides with the midpoint of the hypotenuse; the hypotenuse should be visually emphasized (e.g., slightly bolder stroke) and the right-angle marker at vertex C should be prominent to reinforce the connection.
• For the obtuse triangle case, the perpendicular bisectors should extend noticeably beyond the triangle boundaries so their exterior meeting point is unambiguous. The circumcenter dot should appear clearly outside the triangle with enough space so the circumcircle fits within the main visual area.
• Each triangle should be immediately recognizable as its type: a clearly obtuse angle (e.g., > 120°), a clean right angle marker, and an all-acute scalene shape.
• Midpoint marks, right-angle markers, and radius tick marks should use a consistent visual style across all three cases.
• Transitions between cases use a fade-out/fade-in to clearly separate the three demonstrations.