Geometry Problem – Midpoint and Angle Bisector

Overview

A clean, professional animation that constructs a scalene triangle $ABC$ with given side lengths, marks the midpoint of $BC$ and the internal angle bisector from $A$, then walks through the algebraic steps to find the length $MD$. The left 65 % of the screen continuously shows the triangle, while the right 35 % displays all calculations step‑by‑step.

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Phases

#	Phase Name	Duration	Description
1	Scene 1 – Draw Figure	~6 s	Draw triangle $ABC$ with sides $AB=9$, $AC=7$, $BC=8$. Place red midpoint $M$ on $BC$, draw angle bisector $AD$ intersecting $BC$ at blue point $D$. Add equal angle arcs at $A$.
2	Scene 2 – Midpoint	~5 s	Highlight $M$. Reveal the three equations for the midpoint calculation one after another in the right panel.
3	Scene 3 – Angle‑Bisector Theorem	~5 s	Highlight segment $AD$ and its arcs briefly. Show the proportion $\frac{BD}{DC}=\frac{AB}{AC}$ and then substitute the numeric values.
4	Scene 4 – Find $BD$	~7 s	Display the system $BD+DC=8$ and the derived formula $BD = \frac{9}{9+7}\ imes8$. Compute stepwise to $BD=4.5$. Highlight point $D$.
5	Scene 5 – Find $MD$	~7 s	Highlight both $M$ and $D$; draw segment $MD$ with emphasis. Present the three equations $MD = BD - BM$, substitute numbers, and finish with the result $MD = 0.5$ units displayed prominently.

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Layout

┌───────────────────────────────────────────────────────┐
│                                                       │
│  LEFT (≈65 % width)                                    │
│  Main – triangle ABC with all points and constructions │
│                                                       │
├───────────────────────────────────────────────────────┤
│  RIGHT (≈35 % width)                                   │
│  Calculation panel – sequential equations, each      │
│  appearing one‑by‑one, never overlapping the triangle │
└───────────────────────────────────────────────────────┘

Area Descriptions

Area	Content	Notes
Main	Triangle $ABC$ with side labels, red midpoint $M$, blue point $D$, angle bisector line $AD$, and two small arcs at $A$ indicating equal angles.	Remains visible throughout all phases; points may be highlighted (e.g., change point color or add a halo) when referenced.
Calculation	Textual mathematical expressions appearing sequentially, each centered vertically within the right panel.	No text ever appears over the triangle; each new line fades in while previous lines stay visible unless explicitly replaced by the next step.

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Notes

• Background: Pure white throughout; all geometric lines in black, points as solid circles (red for $M$, blue for $D$).
• Timing: Durations are approximate; each equation is given ~1 s to appear and ~0.5 s to settle before the next.
• Highlighting: When a point or segment is the focus of a phase, it is emphasized by a brief increase in line thickness (or a subtle glow) while the rest stays unchanged.
• No extra text: Only the required equations and the final result are shown; no explanatory sentences or subtitles.
• Assumptions:
  1. The triangle is positioned so that side $BC$ is horizontal at the bottom, making the left‑right split natural.
  2. The angle arcs at $A$ are small enough not to interfere with other elements.
  3. All numeric calculations are performed exactly as given; rounding is not needed beyond the displayed values.
• The specification is designed for a single Manim Scene class.